Theorem vtocl3ga 2796

Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.)

Hypotheses

Ref	Expression
vtocl3ga.1	|- ( x = A -> ( ph <-> ps ) )
vtocl3ga.2	|- ( y = B -> ( ps <-> ch ) )
vtocl3ga.3	|- ( z = C -> ( ch <-> th ) )
vtocl3ga.4	|- ( ( x e. D /\ y e. R /\ z e. S ) -> ph )

Assertion

Ref	Expression
vtocl3ga	|- ( ( A e. D /\ B e. R /\ C e. S ) -> th )

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

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

Proof of Theorem vtocl3ga

Step	Hyp	Ref	Expression
1		nfcv 2308	. 2 |- F/_ x A
2		nfcv 2308	. 2 |- F/_ y A
3		nfcv 2308	. 2 |- F/_ z A
4		nfcv 2308	. 2 |- F/_ y B
5		nfcv 2308	. 2 |- F/_ z B
6		nfcv 2308	. 2 |- F/_ z C
7		nfv 1516	. 2 |- F/ x ps
8		nfv 1516	. 2 |- F/ y ch
9		nfv 1516	. 2 |- F/ z th
10		vtocl3ga.1	. 2 |- ( x = A -> ( ph <-> ps ) )
11		vtocl3ga.2	. 2 |- ( y = B -> ( ps <-> ch ) )
12		vtocl3ga.3	. 2 |- ( z = C -> ( ch <-> th ) )
13		vtocl3ga.4	. 2 |- ( ( x e. D /\ y e. R /\ z e. S ) -> ph )
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	vtocl3gaf 2795	1 |- ( ( A e. D /\ B e. R /\ C e. S ) -> th )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728

This theorem is referenced by:  preq12bg  3753  pocl  4281  sowlin  4298