Metamath Proof Explorer

Theorem vtoclgft

Description: Closed theorem form of vtoclgf . The reverse implication is proven in ceqsal1t . See ceqsalt for a version with x and A disjoint. (Contributed by NM, 17-Feb-2013) (Revised by Mario Carneiro, 12-Oct-2016) (Proof shortened by JJ, 11-Aug-2021) Avoid ax-13 . (Revised by GG, 6-Oct-2023)

Ref Expression
Assertion vtoclgft 
|- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> ps )

Proof

Step Hyp Ref Expression
1 biimp 
 |-  ( ( ph <-> ps ) -> ( ph -> ps ) )
2 1 imim2i 
 |-  ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ph -> ps ) ) )
3 2 alimi 
 |-  ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( x = A -> ( ph -> ps ) ) )
4 spcimgft 
 |-  ( ( ( F/_ x A /\ F/ x ps ) /\ A. x ( x = A -> ( ph -> ps ) ) ) -> ( A e. V -> ( A. x ph -> ps ) ) )
5 3 4 sylan2 
 |-  ( ( ( F/_ x A /\ F/ x ps ) /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( A e. V -> ( A. x ph -> ps ) ) )
6 5 com23 
 |-  ( ( ( F/_ x A /\ F/ x ps ) /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( A. x ph -> ( A e. V -> ps ) ) )
7 6 impr 
 |-  ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) ) -> ( A e. V -> ps ) )
8 7 3impia 
 |-  ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> ps )