Theorem ceqsalg 3230

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. For an alternate proof, see ceqsalgALT 3231. (Contributed by NM, 29-Oct-2003.) (Proof shortened by BJ, 29-Sep-2019.)

Hypotheses

Ref	Expression
ceqsalg.1	|- F/ x ps
ceqsalg.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
ceqsalg	|- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) )

Distinct variable group:   x, A

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

Proof of Theorem ceqsalg

Step	Hyp	Ref	Expression
1		ceqsalg.1	. 2 |- F/ x ps
2		ceqsalg.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	ax-gen 1722	. 2 |- A. x ( x = A -> ( ph <-> ps ) )
4		ceqsalt 3228	. 2 |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( A. x ( x = A -> ph ) <-> ps ) )
5	1, 3, 4	mp3an12 1414	1 |- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) )

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

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-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  ceqsal  3232  uniiunlem  3691  ralrnmpt2  6775  fimaxre3  10970  pmapglbx  35055