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Metamath Proof Explorer

Theorem rspct

Description: A closed version of rspc . (Contributed by Andrew Salmon, 6-Jun-2011)

Ref Expression
Hypothesis rspct.1 
|- F/ x ps
Assertion rspct 
|- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A. x e. B ph -> ps ) ) )

Proof

Step Hyp Ref Expression
1 rspct.1 
 |-  F/ x ps
2 df-ral 
 |-  ( A. x e. B ph <-> A. x ( x e. B -> ph ) )
3 eleq1 
 |-  ( x = A -> ( x e. B <-> A e. B ) )
4 3 adantr 
 |-  ( ( x = A /\ ( ph <-> ps ) ) -> ( x e. B <-> A e. B ) )
5 simpr 
 |-  ( ( x = A /\ ( ph <-> ps ) ) -> ( ph <-> ps ) )
6 4 5 imbi12d 
 |-  ( ( x = A /\ ( ph <-> ps ) ) -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) )
7 6 ex 
 |-  ( x = A -> ( ( ph <-> ps ) -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) ) )
8 7 a2i 
 |-  ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) ) )
9 8 alimi 
 |-  ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( x = A -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) ) )
10 nfv 
 |-  F/ x A e. B
11 10 1 nfim 
 |-  F/ x ( A e. B -> ps )
12 nfcv 
 |-  F/_ x A
13 11 12 spcgft 
 |-  ( A. x ( x = A -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) ) -> ( A e. B -> ( A. x ( x e. B -> ph ) -> ( A e. B -> ps ) ) ) )
14 9 13 syl 
 |-  ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A. x ( x e. B -> ph ) -> ( A e. B -> ps ) ) ) )
15 2 14 syl7bi 
 |-  ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A. x e. B ph -> ( A e. B -> ps ) ) ) )
16 15 com34 
 |-  ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A e. B -> ( A. x e. B ph -> ps ) ) ) )
17 16 pm2.43d 
 |-  ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A. x e. B ph -> ps ) ) )