Theorem vtocl2d 29314

Description: Implicit substitution of two classes for two setvar variables. (Contributed by Thierry Arnoux, 25-Aug-2020.)

Hypotheses

Ref	Expression
vtocl2d.a	|- ( ph -> A e. V )
vtocl2d.b	|- ( ph -> B e. W )
vtocl2d.1	|- ( ( x = A /\ y = B ) -> ( ps <-> ch ) )
vtocl2d.3	|- ( ph -> ps )

Assertion

Ref	Expression
vtocl2d	|- ( ph -> ch )

Distinct variable groups:   x, A, y   x, B, y   x, V   x, W, y   ch, x, y   ph, x, y

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

Proof of Theorem vtocl2d

Step	Hyp	Ref	Expression
1		vtocl2d.b	. . 3 |- ( ph -> B e. W )
2		vtocl2d.a	. . 3 |- ( ph -> A e. V )
3		nfcv 2764	. . . 4 |- F/_ y B
4		nfcv 2764	. . . 4 |- F/_ x B
5		nfcv 2764	. . . 4 |- F/_ x A
6		nfv 1843	. . . . 5 |- F/ y ph
7		nfsbc1v 3455	. . . . 5 |- F/ y [. B / y ]. ps
8	6, 7	nfim 1825	. . . 4 |- F/ y ( ph -> [. B / y ]. ps )
9		nfv 1843	. . . 4 |- F/ x ( ph -> ch )
10		sbceq1a 3446	. . . . 5 |- ( y = B -> ( ps <-> [. B / y ]. ps ) )
11	10	imbi2d 330	. . . 4 |- ( y = B -> ( ( ph -> ps ) <-> ( ph -> [. B / y ]. ps ) ) )
12		sbceq1a 3446	. . . . . 6 |- ( x = A -> ( [. B / y ]. ps <-> [. A / x ]. [. B / y ]. ps ) )
13		nfv 1843	. . . . . . . 8 |- F/ x ch
14		nfv 1843	. . . . . . . 8 |- F/ y ch
15		nfv 1843	. . . . . . . 8 |- F/ x B e. W
16		vtocl2d.1	. . . . . . . 8 |- ( ( x = A /\ y = B ) -> ( ps <-> ch ) )
17	13, 14, 15, 16	sbc2iegf 3504	. . . . . . 7 |- ( ( A e. V /\ B e. W ) -> ( [. A / x ]. [. B / y ]. ps <-> ch ) )
18	2, 1, 17	syl2anc 693	. . . . . 6 |- ( ph -> ( [. A / x ]. [. B / y ]. ps <-> ch ) )
19	12, 18	sylan9bb 736	. . . . 5 |- ( ( x = A /\ ph ) -> ( [. B / y ]. ps <-> ch ) )
20	19	pm5.74da 723	. . . 4 |- ( x = A -> ( ( ph -> [. B / y ]. ps ) <-> ( ph -> ch ) ) )
21		vtocl2d.3	. . . 4 |- ( ph -> ps )
22	3, 4, 5, 8, 9, 11, 20, 21	vtocl2gf 3268	. . 3 |- ( ( B e. W /\ A e. V ) -> ( ph -> ch ) )
23	1, 2, 22	syl2anc 693	. 2 |- ( ph -> ( ph -> ch ) )
24	23	pm2.43i 52	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   [.wsbc 3435

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-3an 1039  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-v 3202  df-sbc 3436

This theorem is referenced by:  submateq  29875