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Metamath Proof Explorer

Theorem spcimgfi1OLD

Description: Obsolete version of spcimgfi1 as of 27-Jul-2025. (Contributed by Mario Carneiro, 4-Jan-2017) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses spcimgfi1.1 
|- F/ x ps
spcimgfi1.2 
|- F/_ x A
Assertion spcimgfi1OLD 
|- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. B -> ( A. x ph -> ps ) ) )

Proof

Step Hyp Ref Expression
1 spcimgfi1.1 
 |-  F/ x ps
2 spcimgfi1.2 
 |-  F/_ x A
3 elex 
 |-  ( A e. B -> A e. _V )
4 2 issetf 
 |-  ( A e. _V <-> E. x x = A )
5 exim 
 |-  ( A. x ( x = A -> ( ph -> ps ) ) -> ( E. x x = A -> E. x ( ph -> ps ) ) )
6 4 5 biimtrid 
 |-  ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. _V -> E. x ( ph -> ps ) ) )
7 1 19.36 
 |-  ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) )
8 6 7 imbitrdi 
 |-  ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. _V -> ( A. x ph -> ps ) ) )
9 3 8 syl5 
 |-  ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. B -> ( A. x ph -> ps ) ) )