Theorem ceqsexg 1411

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis.

Hypotheses

Ref	Expression
ceqsexg.1	|- (ps -> A.xps)
ceqsexg.2	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
ceqsexg	|- (A e. B -> (E.x(x = A /\ ph) <-> ps))

Distinct variable group(s):   x,A

Proof of Theorem ceqsexg

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 |- (y e. A -> A.x y e. A)
2		hbe1 709	. . 3 |- (E.x(x = A /\ ph) -> A.xE.x(x = A /\ ph))
3		ceqsexg.1	. . 3 |- (ps -> A.xps)
4	2, 3	hbbi 705	. 2 |- ((E.x(x = A /\ ph) <-> ps) -> A.x(E.x(x = A /\ ph) <-> ps))
5		ceqex 1410	. . 3 |- (x = A -> (ph <-> E.x(x = A /\ ph)))
6		ceqsexg.2	. . 3 |- (x = A -> (ph <-> ps))
7	5, 6	bibi12d 477	. 2 |- (x = A -> ((ph <-> ph) <-> (E.x(x = A /\ ph) <-> ps)))
8		pm4.2 148	. 2 |- (ph <-> ph)
9	1, 4, 7, 8	vtoclgf 1382	1 |- (A e. B -> (E.x(x = A /\ ph) <-> ps))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = wceq 1091   e. wcel 1092

This theorem is referenced by:  ceqsexgv 1412  sbc5g 1450

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-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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