Theorem rexraleqim 3328

Description: Statement following from existence and generalization with equality. (Contributed by AV, 9-Feb-2019.)

Hypotheses

Ref	Expression
rexraleqim.1	|- ( x = z -> ( ps <-> ph ) )
rexraleqim.2	|- ( z = Y -> ( ph <-> th ) )

Assertion

Ref	Expression
rexraleqim	|- ( ( E. z e. A ph /\ A. x e. A ( ps -> x = Y ) ) -> th )

Distinct variable groups:   x, A, z   x, Y, z   ph, x   ps, z   th, z

Allowed substitution hints:   ph( z)   ps( x)   th( x)

Proof of Theorem rexraleqim

Step	Hyp	Ref	Expression
1		rexraleqim.1	. . . . . . 7 |- ( x = z -> ( ps <-> ph ) )
2		eqeq1 2626	. . . . . . 7 |- ( x = z -> ( x = Y <-> z = Y ) )
3	1, 2	imbi12d 334	. . . . . 6 |- ( x = z -> ( ( ps -> x = Y ) <-> ( ph -> z = Y ) ) )
4	3	rspcva 3307	. . . . 5 |- ( ( z e. A /\ A. x e. A ( ps -> x = Y ) ) -> ( ph -> z = Y ) )
5		rexraleqim.2	. . . . . 6 |- ( z = Y -> ( ph <-> th ) )
6	5	biimpd 219	. . . . 5 |- ( z = Y -> ( ph -> th ) )
7	4, 6	syli 39	. . . 4 |- ( ( z e. A /\ A. x e. A ( ps -> x = Y ) ) -> ( ph -> th ) )
8	7	impancom 456	. . 3 |- ( ( z e. A /\ ph ) -> ( A. x e. A ( ps -> x = Y ) -> th ) )
9	8	rexlimiva 3028	. 2 |- ( E. z e. A ph -> ( A. x e. A ( ps -> x = Y ) -> th ) )
10	9	imp 445	1 |- ( ( E. z e. A ph /\ A. x e. A ( ps -> x = Y ) ) -> th )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202

This theorem is referenced by:  cramerlem3  20495