Theorem bj-ceqsal 32882

Description: Remove from ceqsal 3232 dependency on ax-ext 2602 (and on df-cleq 2615, df-v 3202, df-clab 2609, df-sb 1881). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-ceqsal.1	|- F/ x ps
bj-ceqsal.2	|- A e. _V
bj-ceqsal.3	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
bj-ceqsal	|- ( A. x ( x = A -> ph ) <-> ps )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem bj-ceqsal

Step	Hyp	Ref	Expression
1		bj-ceqsal.2	. 2 |- A e. _V
2		bj-ceqsal.1	. . 3 |- F/ x ps
3		bj-ceqsal.3	. . 3 |- ( x = A -> ( ph <-> ps ) )
4	2, 3	bj-ceqsalgv 32880	. 2 |- ( A e. _V -> ( A. x ( x = A -> ph ) <-> ps ) )
5	1, 4	ax-mp 5	1 |- ( A. x ( x = A -> ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   F/wnf 1708   e. wcel 1990   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-ex 1705  df-nf 1710  df-clel 2618

This theorem is referenced by:  bj-ceqsalv  32883