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Metamath Proof Explorer

Theorem ceqsralt

Description: Restricted quantifier version of ceqsalt . (Contributed by NM, 28-Feb-2013) (Revised by Mario Carneiro, 10-Oct-2016)

Ref Expression
Assertion ceqsralt 
|- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A. x e. B ( x = A -> ph ) <-> ps ) )

Proof

Step Hyp Ref Expression
1 biimt 
 |-  ( A e. B -> ( A. x ( x = A -> ph ) <-> ( A e. B -> A. x ( x = A -> ph ) ) ) )
2 df-ral 
 |-  ( A. x e. B ( x = A -> ph ) <-> A. x ( x e. B -> ( x = A -> ph ) ) )
3 eleq1 
 |-  ( x = A -> ( x e. B <-> A e. B ) )
4 3 pm5.32ri 
 |-  ( ( x e. B /\ x = A ) <-> ( A e. B /\ x = A ) )
5 4 imbi1i 
 |-  ( ( ( x e. B /\ x = A ) -> ph ) <-> ( ( A e. B /\ x = A ) -> ph ) )
6 impexp 
 |-  ( ( ( x e. B /\ x = A ) -> ph ) <-> ( x e. B -> ( x = A -> ph ) ) )
7 impexp 
 |-  ( ( ( A e. B /\ x = A ) -> ph ) <-> ( A e. B -> ( x = A -> ph ) ) )
8 5 6 7 3bitr3i 
 |-  ( ( x e. B -> ( x = A -> ph ) ) <-> ( A e. B -> ( x = A -> ph ) ) )
9 8 albii 
 |-  ( A. x ( x e. B -> ( x = A -> ph ) ) <-> A. x ( A e. B -> ( x = A -> ph ) ) )
10 19.21v 
 |-  ( A. x ( A e. B -> ( x = A -> ph ) ) <-> ( A e. B -> A. x ( x = A -> ph ) ) )
11 2 9 10 3bitrri 
 |-  ( ( A e. B -> A. x ( x = A -> ph ) ) <-> A. x e. B ( x = A -> ph ) )
12 1 11 bitrdi 
 |-  ( A e. B -> ( A. x ( x = A -> ph ) <-> A. x e. B ( x = A -> ph ) ) )
13 12 3ad2ant3 
 |-  ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A. x ( x = A -> ph ) <-> A. x e. B ( x = A -> ph ) ) )
14 ceqsalt 
 |-  ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A. x ( x = A -> ph ) <-> ps ) )
15 13 14 bitr3d 
 |-  ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A. x e. B ( x = A -> ph ) <-> ps ) )