Theorem ax11ev 1709

Description: Analogue to ax11v 1708 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 1586	. 2 |- E. z z = y
2		ax11e 1677	. . . . 5 |- ( x = z -> ( E. x ( x = z /\ ph ) -> E. z ph ) )
3		ax-17 1419	. . . . . 6 |- ( ph -> A. z ph )
4	3	19.9h 1534	. . . . 5 |- ( E. z ph <-> ph )
5	2, 4	syl6ib 150	. . . 4 |- ( x = z -> ( E. x ( x = z /\ ph ) -> ph ) )
6		equequ2 1599	. . . . 5 |- ( z = y -> ( x = z <-> x = y ) )
7	6	anbi1d 438	. . . . . . 7 |- ( z = y -> ( ( x = z /\ ph ) <-> ( x = y /\ ph ) ) )
8	7	exbidv 1706	. . . . . 6 |- ( z = y -> ( E. x ( x = z /\ ph ) <-> E. x ( x = y /\ ph ) ) )
9	8	imbi1d 220	. . . . 5 |- ( z = y -> ( ( E. x ( x = z /\ ph ) -> ph ) <-> ( E. x ( x = y /\ ph ) -> ph ) ) )
10	6, 9	imbi12d 223	. . . 4 |- ( z = y -> ( ( x = z -> ( E. x ( x = z /\ ph ) -> ph ) ) <-> ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) ) ) )
11	5, 10	mpbii 136	. . 3 |- ( z = y -> ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) ) )
12	11	exlimiv 1489	. 2 |- ( E. z z = y -> ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) ) )
13	1, 12	ax-mp 7	1 |- ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)