Theorem 2gencl 2784

Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)

Hypotheses

Ref	Expression
2gencl.1	|- ( C e. S <-> E. x e. R A = C )
2gencl.2	|- ( D e. S <-> E. y e. R B = D )
2gencl.3	|- ( A = C -> ( ph <-> ps ) )
2gencl.4	|- ( B = D -> ( ps <-> ch ) )
2gencl.5	|- ( ( x e. R /\ y e. R ) -> ph )

Assertion

Ref	Expression
2gencl	|- ( ( C e. S /\ D e. S ) -> ch )

Distinct variable groups:   x, y   x, R   ps, x   y, C   y, S   ch, y

Allowed substitution hints:   ph( x, y)   ps( y)   ch( x)   A( x, y)   B( x, y)   C( x)   D( x, y)   R( y)   S( x)

Proof of Theorem 2gencl

Step	Hyp	Ref	Expression
1		2gencl.2	. . . 4 |- ( D e. S <-> E. y e. R B = D )
2		df-rex 2473	. . . 4 |- ( E. y e. R B = D <-> E. y ( y e. R /\ B = D ) )
3	1, 2	bitri 184	. . 3 |- ( D e. S <-> E. y ( y e. R /\ B = D ) )
4		2gencl.4	. . . 4 |- ( B = D -> ( ps <-> ch ) )
5	4	imbi2d 230	. . 3 |- ( B = D -> ( ( C e. S -> ps ) <-> ( C e. S -> ch ) ) )
6		2gencl.1	. . . . . 6 |- ( C e. S <-> E. x e. R A = C )
7		df-rex 2473	. . . . . 6 |- ( E. x e. R A = C <-> E. x ( x e. R /\ A = C ) )
8	6, 7	bitri 184	. . . . 5 |- ( C e. S <-> E. x ( x e. R /\ A = C ) )
9		2gencl.3	. . . . . 6 |- ( A = C -> ( ph <-> ps ) )
10	9	imbi2d 230	. . . . 5 |- ( A = C -> ( ( y e. R -> ph ) <-> ( y e. R -> ps ) ) )
11		2gencl.5	. . . . . 6 |- ( ( x e. R /\ y e. R ) -> ph )
12	11	ex 115	. . . . 5 |- ( x e. R -> ( y e. R -> ph ) )
13	8, 10, 12	gencl 2783	. . . 4 |- ( C e. S -> ( y e. R -> ps ) )
14	13	com12 30	. . 3 |- ( y e. R -> ( C e. S -> ps ) )
15	3, 5, 14	gencl 2783	. 2 |- ( D e. S -> ( C e. S -> ch ) )
16	15	impcom 125	1 |- ( ( C e. S /\ D e. S ) -> ch )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1363   E.wex 1502   e. wcel 2159   E.wrex 2468

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-gen 1459  ax-ie2 1504  ax-17 1536

This theorem depends on definitions:  df-bi 117  df-rex 2473

This theorem is referenced by:  3gencl  2785  axaddrcl  7881  axmulrcl  7883  axpre-apti  7901  axpre-mulgt0  7903  uzin2  11013