Theorem 2gencl 2889

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 2621	. . . 4 |- (E.y e. R B = D <-> E.y(y e. R /\ B = D))
3	1, 2	bitri 240	. . 3 |- (D e. S <-> E.y(y e. R /\ B = D))
4		2gencl.4	. . . 4 |- (B = D -> (ps <-> ch))
5	4	imbi2d 307	. . 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 2621	. . . . . 6 |- (E.x e. R A = C <-> E.x(x e. R /\ A = C))
8	6, 7	bitri 240	. . . . 5 |- (C e. S <-> E.x(x e. R /\ A = C))
9		2gencl.3	. . . . . 6 |- (A = C -> (ph <-> ps))
10	9	imbi2d 307	. . . . 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 423	. . . . 5 |- (x e. R -> (y e. R -> ph))
13	8, 10, 12	gencl 2888	. . . 4 |- (C e. S -> (y e. R -> ps))
14	13	com12 27	. . 3 |- (y e. R -> (C e. S -> ps))
15	3, 5, 14	gencl 2888	. 2 |- (D e. S -> (C e. S -> ch))
16	15	impcom 419	1 |- ((C e. S /\ D e. S) -> ch)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-rex 2621

This theorem is referenced by:  3gencl  2890