Theorem ax11ev 1808

Description: Analogue to ax11v 1807 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)

Assertion

Ref	Expression
ax11ev	|- ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem ax11ev

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		a9e 1676	. 2 |- E. z z = y
2		ax11e 1776	. . . . 5 |- ( x = z -> ( E. x ( x = z /\ ph ) -> E. z ph ) )
3		ax-17 1506	. . . . . 6 |- ( ph -> A. z ph )
4	3	19.9h 1623	. . . . 5 |- ( E. z ph <-> ph )
5	2, 4	syl6ib 160	. . . 4 |- ( x = z -> ( E. x ( x = z /\ ph ) -> ph ) )
6		equequ2 1693	. . . . 5 |- ( z = y -> ( x = z <-> x = y ) )
7	6	anbi1d 461	. . . . . . 7 |- ( z = y -> ( ( x = z /\ ph ) <-> ( x = y /\ ph ) ) )
8	7	exbidv 1805	. . . . . 6 |- ( z = y -> ( E. x ( x = z /\ ph ) <-> E. x ( x = y /\ ph ) ) )
9	8	imbi1d 230	. . . . 5 |- ( z = y -> ( ( E. x ( x = z /\ ph ) -> ph ) <-> ( E. x ( x = y /\ ph ) -> ph ) ) )
10	6, 9	imbi12d 233	. . . 4 |- ( z = y -> ( ( x = z -> ( E. x ( x = z /\ ph ) -> ph ) ) <-> ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) ) ) )
11	5, 10	mpbii 147	. . 3 |- ( z = y -> ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) ) )
12	11	exlimiv 1578	. 2 |- ( E. z z = y -> ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) ) )
13	1, 12	ax-mp 5	1 |- ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   E.wex 1472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)