Theorem alexeqg 3332

Description: Two ways to express substitution of A for x in ph. This is the analogue for classes of sb56 2150. (Contributed by NM, 2-Mar-1995.) (Revised by BJ, 27-Apr-2019.)

Assertion

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

Distinct variable group:   x, A

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

Proof of Theorem alexeqg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2633	. . . . 5 |- ( y = A -> ( x = y <-> x = A ) )
2	1	anbi1d 741	. . . 4 |- ( y = A -> ( ( x = y /\ ph ) <-> ( x = A /\ ph ) ) )
3	2	exbidv 1850	. . 3 |- ( y = A -> ( E. x ( x = y /\ ph ) <-> E. x ( x = A /\ ph ) ) )
4	1	imbi1d 331	. . . 4 |- ( y = A -> ( ( x = y -> ph ) <-> ( x = A -> ph ) ) )
5	4	albidv 1849	. . 3 |- ( y = A -> ( A. x ( x = y -> ph ) <-> A. x ( x = A -> ph ) ) )
6		sb56 2150	. . 3 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
7	3, 5, 6	vtoclbg 3267	. 2 |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> A. x ( x = A -> ph ) ) )
8	7	bicomd 213	1 |- ( A e. V -> ( A. x ( 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

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:  ceqex  3333  sbc6g  3461