Theorem spc2ed 6236

Description: Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)

Hypotheses

Ref	Expression
spc2ed.x	|- F/ x ch
spc2ed.y	|- F/ y ch
spc2ed.1	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
spc2ed	|- ( ( ph /\ ( A e. V /\ B e. W ) ) -> ( ch -> E. x E. y ps ) )

Distinct variable groups:   x, y, A   x, B, y   ph, x, y

Allowed substitution hints:   ps( x, y)   ch( x, y)   V( x, y)   W( x, y)

Proof of Theorem spc2ed

Step	Hyp	Ref	Expression
1		elisset 2753	. . . 4 |- ( A e. V -> E. x x = A )
2		elisset 2753	. . . 4 |- ( B e. W -> E. y y = B )
3	1, 2	anim12i 338	. . 3 |- ( ( A e. V /\ B e. W ) -> ( E. x x = A /\ E. y y = B ) )
4		eeanv 1932	. . 3 |- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
5	3, 4	sylibr 134	. 2 |- ( ( A e. V /\ B e. W ) -> E. x E. y ( x = A /\ y = B ) )
6		nfv 1528	. . . . 5 |- F/ x ph
7		spc2ed.x	. . . . 5 |- F/ x ch
8	6, 7	nfan 1565	. . . 4 |- F/ x ( ph /\ ch )
9		nfv 1528	. . . . . 6 |- F/ y ph
10		spc2ed.y	. . . . . 6 |- F/ y ch
11	9, 10	nfan 1565	. . . . 5 |- F/ y ( ph /\ ch )
12		anass 401	. . . . . . . 8 |- ( ( ( ch /\ ph ) /\ ( x = A /\ y = B ) ) <-> ( ch /\ ( ph /\ ( x = A /\ y = B ) ) ) )
13		ancom 266	. . . . . . . . 9 |- ( ( ch /\ ph ) <-> ( ph /\ ch ) )
14	13	anbi1i 458	. . . . . . . 8 |- ( ( ( ch /\ ph ) /\ ( x = A /\ y = B ) ) <-> ( ( ph /\ ch ) /\ ( x = A /\ y = B ) ) )
15	12, 14	bitr3i 186	. . . . . . 7 |- ( ( ch /\ ( ph /\ ( x = A /\ y = B ) ) ) <-> ( ( ph /\ ch ) /\ ( x = A /\ y = B ) ) )
16		spc2ed.1	. . . . . . . 8 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) )
17	16	biimparc 299	. . . . . . 7 |- ( ( ch /\ ( ph /\ ( x = A /\ y = B ) ) ) -> ps )
18	15, 17	sylbir 135	. . . . . 6 |- ( ( ( ph /\ ch ) /\ ( x = A /\ y = B ) ) -> ps )
19	18	ex 115	. . . . 5 |- ( ( ph /\ ch ) -> ( ( x = A /\ y = B ) -> ps ) )
20	11, 19	eximd 1612	. . . 4 |- ( ( ph /\ ch ) -> ( E. y ( x = A /\ y = B ) -> E. y ps ) )
21	8, 20	eximd 1612	. . 3 |- ( ( ph /\ ch ) -> ( E. x E. y ( x = A /\ y = B ) -> E. x E. y ps ) )
22	21	impancom 260	. 2 |- ( ( ph /\ E. x E. y ( x = A /\ y = B ) ) -> ( ch -> E. x E. y ps ) )
23	5, 22	sylan2 286	1 |- ( ( ph /\ ( A e. V /\ B e. W ) ) -> ( ch -> E. x E. y ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   F/wnf 1460   E.wex 1492   e. wcel 2148

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2741

This theorem is referenced by:  cnvoprab  6237