Theorem ceqex 3333

Description: Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.) (Proof shortened by BJ, 1-May-2019.)

Assertion

Ref	Expression
ceqex	|- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem ceqex

Step	Hyp	Ref	Expression
1		19.8a 2052	. . 3 |- ( ( x = A /\ ph ) -> E. x ( x = A /\ ph ) )
2	1	ex 450	. 2 |- ( x = A -> ( ph -> E. x ( x = A /\ ph ) ) )
3		eqvisset 3211	. . . 4 |- ( x = A -> A e. _V )
4		alexeqg 3332	. . . 4 |- ( A e. _V -> ( A. x ( x = A -> ph ) <-> E. x ( x = A /\ ph ) ) )
5	3, 4	syl 17	. . 3 |- ( x = A -> ( A. x ( x = A -> ph ) <-> E. x ( x = A /\ ph ) ) )
6		sp 2053	. . . 4 |- ( A. x ( x = A -> ph ) -> ( x = A -> ph ) )
7	6	com12 32	. . 3 |- ( x = A -> ( A. x ( x = A -> ph ) -> ph ) )
8	5, 7	sylbird 250	. 2 |- ( x = A -> ( E. x ( x = A /\ ph ) -> ph ) )
9	2, 8	impbid 202	1 |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   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-9 1999  ax-10 2019  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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:  ceqsexg  3334