Theorem gencl 2586

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

Hypotheses

Ref	Expression
gencl.1	|- ( th <-> E. x ( ch /\ A = B ) )
gencl.2	|- ( A = B -> ( ph <-> ps ) )
gencl.3	|- ( ch -> ph )

Assertion

Ref	Expression
gencl	|- ( th -> ps )

Distinct variable group:   ps, x

Allowed substitution hints:   ph( x)   ch( x)   th( x)   A( x)   B( x)

Proof of Theorem gencl

Step	Hyp	Ref	Expression
1		gencl.1	. 2 |- ( th <-> E. x ( ch /\ A = B ) )
2		gencl.3	. . . . 5 |- ( ch -> ph )
3		gencl.2	. . . . 5 |- ( A = B -> ( ph <-> ps ) )
4	2, 3	syl5ib 143	. . . 4 |- ( A = B -> ( ch -> ps ) )
5	4	impcom 116	. . 3 |- ( ( ch /\ A = B ) -> ps )
6	5	exlimiv 1489	. 2 |- ( E. x ( ch /\ A = B ) -> ps )
7	1, 6	sylbi 114	1 |- ( th -> ps )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-gen 1338  ax-ie2 1383  ax-17 1419

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  2gencl  2587  3gencl  2588  axprecex  6954  axpre-ltirr  6956