Theorem vtocl4ga 2847

Description: Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.) (Proof shortened by Wolf Lammen, 31-May-2025.)

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		vtocl4ga.3	. . . 4 |- ( z = C -> ( ch <-> rh ) )
2	1	imbi2d 230	. . 3 |- ( z = C -> ( ( ( A e. Q /\ B e. R ) -> ch ) <-> ( ( A e. Q /\ B e. R ) -> rh ) ) )
3		vtocl4ga.4	. . . 4 |- ( w = D -> ( rh <-> th ) )
4	3	imbi2d 230	. . 3 |- ( w = D -> ( ( ( A e. Q /\ B e. R ) -> rh ) <-> ( ( A e. Q /\ B e. R ) -> th ) ) )
5		vtocl4ga.1	. . . . . 6 |- ( x = A -> ( ph <-> ps ) )
6	5	imbi2d 230	. . . . 5 |- ( x = A -> ( ( ( z e. S /\ w e. T ) -> ph ) <-> ( ( z e. S /\ w e. T ) -> ps ) ) )
7		vtocl4ga.2	. . . . . 6 |- ( y = B -> ( ps <-> ch ) )
8	7	imbi2d 230	. . . . 5 |- ( y = B -> ( ( ( z e. S /\ w e. T ) -> ps ) <-> ( ( z e. S /\ w e. T ) -> ch ) ) )
9		vtocl4ga.5	. . . . . 6 |- ( ( ( x e. Q /\ y e. R ) /\ ( z e. S /\ w e. T ) ) -> ph )
10	9	ex 115	. . . . 5 |- ( ( x e. Q /\ y e. R ) -> ( ( z e. S /\ w e. T ) -> ph ) )
11	6, 8, 10	vtocl2ga 2843	. . . 4 |- ( ( A e. Q /\ B e. R ) -> ( ( z e. S /\ w e. T ) -> ch ) )
12	11	com12 30	. . 3 |- ( ( z e. S /\ w e. T ) -> ( ( A e. Q /\ B e. R ) -> ch ) )
13	2, 4, 12	vtocl2ga 2843	. 2 |- ( ( C e. S /\ D e. T ) -> ( ( A e. Q /\ B e. R ) -> th ) )
14	13	impcom 125	1 |- ( ( ( A e. Q /\ B e. R ) /\ ( C e. S /\ D e. T ) ) -> th )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2177

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775

This theorem is referenced by:  wrd2ind  11194