Theorem vtocl4g 2846

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

Hypotheses

Ref	Expression
vtocl4g.1	|- ( x = A -> ( ph <-> ps ) )
vtocl4g.2	|- ( y = B -> ( ps <-> ch ) )
vtocl4g.3	|- ( z = C -> ( ch <-> rh ) )
vtocl4g.4	|- ( w = D -> ( rh <-> th ) )
vtocl4g.5	|- ph

Assertion

Ref	Expression
vtocl4g	|- ( ( ( A e. Q /\ B e. R ) /\ ( C e. S /\ D e. T ) ) -> th )

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

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)   Q( x, y)   R( x, y)   S( x, y, z, w)   T( x, y, z, w)

Proof of Theorem vtocl4g

Step	Hyp	Ref	Expression
1		vtocl4g.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		vtocl4g.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		vtocl4g.1	. . . 4 |- ( x = A -> ( ph <-> ps ) )
6		vtocl4g.2	. . . 4 |- ( y = B -> ( ps <-> ch ) )
7		vtocl4g.5	. . . 4 |- ph
8	5, 6, 7	vtocl2g 2839	. . 3 |- ( ( A e. Q /\ B e. R ) -> ch )
9	2, 4, 8	vtocl2g 2839	. 2 |- ( ( C e. S /\ D e. T ) -> ( ( A e. Q /\ B e. R ) -> th ) )
10	9	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: (None)