Metamath Proof Explorer


Theorem exgen

Description: Rule of existential generalization, similar to universal generalization ax-gen , but valid only if an individual exists. Its proof requires ax-6 in our axiomatization but the equality predicate does not occur in its statement. Some fundamental theorems of predicate calculus can be proven from ax-gen , ax-4 and this theorem alone, not requiring ax-7 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017) (Proof shortened by Wolf Lammen, 20-Oct-2023)

Ref Expression
Hypothesis exgen.1
|- ph
Assertion exgen
|- E. x ph

Proof

Step Hyp Ref Expression
1 exgen.1
 |-  ph
2 idd
 |-  ( x = y -> ( ph -> ph ) )
3 2 1 speiv
 |-  E. x ph