Theorem gencl 2318

Description: Implicit substitution for class with embedded variable.

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

Proof of Theorem gencl

Step	Hyp	Ref	Expression
1		gencl.1	. 2 |- (th <-> E.x(ch /\ A = B))
2		gencl.2	. . . . 5 |- (A = B -> (ph <-> ps))
3		gencl.3	. . . . 5 |- (ch -> ph)
4	2, 3	syl5bi 225	. . . 4 |- (A = B -> (ch -> ps))
5	4	impcom 378	. . 3 |- ((ch /\ A = B) -> ps)
6	5	19.23aiv 1674	. 2 |- (E.x(ch /\ A = B) -> ps)
7	1, 6	sylbi 216	1 |- (th -> ps)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298  E.wex 1326

This theorem is referenced by:  2gencl 2319  3gencl 2320  indpi 6186  axrnegex 6436  axrrecex 6437

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327

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