This rule can be used forwards to derive the sub formula of a FOR ALL formula with the variable replaced by another term and backwards to derive the for all formula from which the sub formula can be eliminated.

To use it forwards:

1. select both an empty line and For All Elimination.
2. then select the FOR ALL formula line you wish to use in the elimination (the formula must be in the scope of the empty line).
3. If there is more than one FOR ALL you can choose to eliminate one or all.
• If you choose to eliminate one then you need to enter the term you want replace for the first quantifier variable when prompted. This term must already be in the signature of the empty line.
• If you choose to eliminate all then you need to enter terms one by one to replace all the quantifiers when prompted. These terms must already be in the signature of the empty line.
4. the sub formula will then be added to your proof as a new line with the FOR ALL variables eliminated replaced by the terms you entered.

To use it backwards:

1. select both the goal line and For All Elimination.
2. then enter the For All formula you wish to eliminate from. (Note the For All formula should be one or more For All quantifiers around the goal formula.)
3. The entered formula would be added as the new goal.