Theorem ax11e 1717

Description: Analogue to ax-11 1437 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.)

Assertion

Ref	Expression
ax11e	|- ( x = y -> ( E. x ( x = y /\ ph ) -> E. y ph ) )

Proof of Theorem ax11e

Step	Hyp	Ref	Expression
1		equs5e 1716	. . 3 |- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) )
2	1	19.21bi 1490	. 2 |- ( E. x ( x = y /\ ph ) -> ( x = y -> E. y ph ) )
3	2	com12 30	1 |- ( x = y -> ( E. x ( x = y /\ ph ) -> E. y ph ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-11 1437  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  ax10oe  1718  drex1  1719  sbcof2  1731  ax11ev  1749