Theorem ax11e 1818

Description: Analogue to ax-11 1528 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 1817	. . 3 |- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) )
2	1	19.21bi 1580	. 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 104   = wceq 1372   E.wex 1514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-11 1528  ax-4 1532  ax-ial 1556

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  ax10oe  1819  drex1  1820  sbcof2  1832  ax11ev  1850