Theorem ceqsralv2 31607

Description: Alternate elimination of a restricted universal quantifier, using implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020.)

Hypothesis

Ref	Expression
ceqsralv2.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
ceqsralv2	|- ( A. x e. B ( x = A -> ph ) <-> ( A e. B -> ps ) )

Distinct variable groups:   x, A   x, B   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem ceqsralv2

Step	Hyp	Ref	Expression
1		ceqsralv2.1	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
2	1	notbid 308	. . . 4 |- ( x = A -> ( -. ph <-> -. ps ) )
3	2	ceqsrexv2 31605	. . 3 |- ( E. x e. B ( x = A /\ -. ph ) <-> ( A e. B /\ -. ps ) )
4		rexanali 2998	. . 3 |- ( E. x e. B ( x = A /\ -. ph ) <-> -. A. x e. B ( x = A -> ph ) )
5		annim 441	. . 3 |- ( ( A e. B /\ -. ps ) <-> -. ( A e. B -> ps ) )
6	3, 4, 5	3bitr3i 290	. 2 |- ( -. A. x e. B ( x = A -> ph ) <-> -. ( A e. B -> ps ) )
7	6	con4bii 311	1 |- ( A. x e. B ( x = A -> ph ) <-> ( A e. B -> ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913

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-ral 2917  df-rex 2918  df-v 3202

This theorem is referenced by: (None)