Theorem gencl 2888

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 210	. . . 4 |- (A = B -> (ch -> ps))
5	4	impcom 419	. . 3 |- ((ch /\ A = B) -> ps)
6	5	exlimiv 1634	. 2 |- (E.x(ch /\ A = B) -> ps)
7	1, 6	sylbi 187	1 |- (th -> ps)

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

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

This theorem is referenced by:  2gencl  2889  3gencl  2890