Theorem vtocl4ga 3278

Description: Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.)

Hypotheses

Ref	Expression
vtocl4ga.1	|- ( x = A -> ( ph <-> ps ) )
vtocl4ga.2	|- ( y = B -> ( ps <-> ch ) )
vtocl4ga.3	|- ( z = C -> ( ch <-> rh ) )
vtocl4ga.4	|- ( w = D -> ( rh <-> th ) )
vtocl4ga.5	|- ( ( ( x e. Q /\ y e. R ) /\ ( z e. S /\ w e. T ) ) -> ph )

Assertion

Ref	Expression
vtocl4ga	|- ( ( ( A e. Q /\ B e. R ) /\ ( C e. S /\ D e. T ) ) -> th )

Distinct variable groups:   x, w, y, z, A   w, B, y, z   w, C, z   w, D   w, R, x, y, z   w, S, x, y, z   w, T, x, y, z   w, Q, x, y, z   ps, x   rh, z   th, w   ch, y

Allowed substitution hints:   ph( x, y, z, w)   ps( y, z, w)   ch( x, z, w)   th( x, y, z)   rh( x, y, w)   B( x)   C( x, y)   D( x, y, z)

Proof of Theorem vtocl4ga

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . . . 6 |- ( x = A -> ( x e. Q <-> A e. Q ) )
2	1	anbi1d 741	. . . . 5 |- ( x = A -> ( ( x e. Q /\ y e. R ) <-> ( A e. Q /\ y e. R ) ) )
3	2	anbi1d 741	. . . 4 |- ( x = A -> ( ( ( x e. Q /\ y e. R ) /\ ( z e. S /\ w e. T ) ) <-> ( ( A e. Q /\ y e. R ) /\ ( z e. S /\ w e. T ) ) ) )
4		vtocl4ga.1	. . . 4 |- ( x = A -> ( ph <-> ps ) )
5	3, 4	imbi12d 334	. . 3 |- ( x = A -> ( ( ( ( x e. Q /\ y e. R ) /\ ( z e. S /\ w e. T ) ) -> ph ) <-> ( ( ( A e. Q /\ y e. R ) /\ ( z e. S /\ w e. T ) ) -> ps ) ) )
6		eleq1 2689	. . . . . 6 |- ( y = B -> ( y e. R <-> B e. R ) )
7	6	anbi2d 740	. . . . 5 |- ( y = B -> ( ( A e. Q /\ y e. R ) <-> ( A e. Q /\ B e. R ) ) )
8	7	anbi1d 741	. . . 4 |- ( y = B -> ( ( ( A e. Q /\ y e. R ) /\ ( z e. S /\ w e. T ) ) <-> ( ( A e. Q /\ B e. R ) /\ ( z e. S /\ w e. T ) ) ) )
9		vtocl4ga.2	. . . 4 |- ( y = B -> ( ps <-> ch ) )
10	8, 9	imbi12d 334	. . 3 |- ( y = B -> ( ( ( ( A e. Q /\ y e. R ) /\ ( z e. S /\ w e. T ) ) -> ps ) <-> ( ( ( A e. Q /\ B e. R ) /\ ( z e. S /\ w e. T ) ) -> ch ) ) )
11		eleq1 2689	. . . . . 6 |- ( z = C -> ( z e. S <-> C e. S ) )
12	11	anbi1d 741	. . . . 5 |- ( z = C -> ( ( z e. S /\ w e. T ) <-> ( C e. S /\ w e. T ) ) )
13	12	anbi2d 740	. . . 4 |- ( z = C -> ( ( ( A e. Q /\ B e. R ) /\ ( z e. S /\ w e. T ) ) <-> ( ( A e. Q /\ B e. R ) /\ ( C e. S /\ w e. T ) ) ) )
14		vtocl4ga.3	. . . 4 |- ( z = C -> ( ch <-> rh ) )
15	13, 14	imbi12d 334	. . 3 |- ( z = C -> ( ( ( ( A e. Q /\ B e. R ) /\ ( z e. S /\ w e. T ) ) -> ch ) <-> ( ( ( A e. Q /\ B e. R ) /\ ( C e. S /\ w e. T ) ) -> rh ) ) )
16		eleq1 2689	. . . . . 6 |- ( w = D -> ( w e. T <-> D e. T ) )
17	16	anbi2d 740	. . . . 5 |- ( w = D -> ( ( C e. S /\ w e. T ) <-> ( C e. S /\ D e. T ) ) )
18	17	anbi2d 740	. . . 4 |- ( w = D -> ( ( ( A e. Q /\ B e. R ) /\ ( C e. S /\ w e. T ) ) <-> ( ( A e. Q /\ B e. R ) /\ ( C e. S /\ D e. T ) ) ) )
19		vtocl4ga.4	. . . 4 |- ( w = D -> ( rh <-> th ) )
20	18, 19	imbi12d 334	. . 3 |- ( w = D -> ( ( ( ( A e. Q /\ B e. R ) /\ ( C e. S /\ w e. T ) ) -> rh ) <-> ( ( ( A e. Q /\ B e. R ) /\ ( C e. S /\ D e. T ) ) -> th ) ) )
21		vtocl4ga.5	. . 3 |- ( ( ( x e. Q /\ y e. R ) /\ ( z e. S /\ w e. T ) ) -> ph )
22	5, 10, 15, 20, 21	vtocl4g 3277	. 2 |- ( ( ( A e. Q /\ B e. R ) /\ ( C e. S /\ D e. T ) ) -> ( ( ( A e. Q /\ B e. R ) /\ ( C e. S /\ D e. T ) ) -> th ) )
23	22	pm2.43i 52	1 |- ( ( ( A e. Q /\ B e. R ) /\ ( C e. S /\ D e. T ) ) -> th )

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

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-v 3202

This theorem is referenced by:  wrd2ind  13477