Theorem 2gencl 1832

Description: Implicit substitution for class with embedded variable.

Hypotheses

Ref	Expression
2gencl.1	|- (C e. S <-> E.x(x e. R /\ A = C))
2gencl.2	|- (D e. S <-> E.y(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

Proof of Theorem 2gencl

Step	Hyp	Ref	Expression
1		2gencl.2	. . 3 |- (D e. S <-> E.y(y e. R /\ B = D))
2		2gencl.4	. . . 4 |- (B = D -> (ps <-> ch))
3	2	imbi2d 614	. . 3 |- (B = D -> ((C e. S -> ps) <-> (C e. S -> ch)))
4		2gencl.1	. . . . 5 |- (C e. S <-> E.x(x e. R /\ A = C))
5		2gencl.3	. . . . . 6 |- (A = C -> (ph <-> ps))
6	5	imbi2d 614	. . . . 5 |- (A = C -> ((y e. R -> ph) <-> (y e. R -> ps)))
7		2gencl.5	. . . . . 6 |- ((x e. R /\ y e. R) -> ph)
8	7	ex 373	. . . . 5 |- (x e. R -> (y e. R -> ph))
9	4, 6, 8	gencl 1831	. . . 4 |- (C e. S -> (y e. R -> ps))
10	9	com12 11	. . 3 |- (y e. R -> (C e. S -> ps))
11	1, 3, 10	gencl 1831	. 2 |- (D e. S -> (C e. S -> ch))
12	11	impcom 351	1 |- ((C e. S /\ D e. S) -> ch)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982

This theorem is referenced by:  3gencl 1833  axaddrcl 5284  axmulrcl 5286  pre-axmulgt0 5302

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983

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