Theorem gencl 1365

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(s):   ps,x

Proof of Theorem gencl

Step	Hyp	Ref	Expression
1		gencl.1	. 2 |- (th <-> E.x(ch /\ A = B))
2		gencl.2	. . . . . 6 |- (A = B -> (ph <-> ps))
3		gencl.3	. . . . . 6 |- (ch -> ph)
4	2, 3	syl5bi 183	. . . . 5 |- (A = B -> (ch -> ps))
5	4	com12 13	. . . 4 |- (ch -> (A = B -> ps))
6	5	imp 277	. . 3 |- ((ch /\ A = B) -> ps)
7	6	19.23aiv 952	. 2 |- (E.x(ch /\ A = B) -> ps)
8	1, 7	sylbi 174	1 |- (th -> ps)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091

This theorem is referenced by:  2gencl 1366  3gencl 1367  indpi 3828  axrnegex 4080  axrrecex 4081

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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