Metamath Proof Explorer

Theorem vtoclgftOLD

Description: Obsolete version of vtoclgft as of 6-Oct-2023. (Contributed by NM, 17-Feb-2013) (Revised by Mario Carneiro, 12-Oct-2016) (Proof shortened by JJ, 11-Aug-2021) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion vtoclgftOLD 
|- ( ( ( 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 elisset 
 |-  ( A e. V -> E. z z = A )
2 nfnfc1 
 |-  F/ x F/_ x A
3 nfcvd 
 |-  ( F/_ x A -> F/_ x z )
4 id 
 |-  ( F/_ x A -> F/_ x A )
5 3 4 nfeqd 
 |-  ( F/_ x A -> F/ x z = A )
6 eqeq1 
 |-  ( z = x -> ( z = A <-> x = A ) )
7 6 a1i 
 |-  ( F/_ x A -> ( z = x -> ( z = A <-> x = A ) ) )
8 2 5 7 cbvexd 
 |-  ( F/_ x A -> ( E. z z = A <-> E. x x = A ) )
9 1 8 syl5ib 
 |-  ( F/_ x A -> ( A e. V -> E. x x = A ) )
10 9 ad2antrr 
 |-  ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) ) -> ( A e. V -> E. x x = A ) )
11 10 3impia 
 |-  ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> E. x x = A )
12 biimp 
 |-  ( ( ph <-> ps ) -> ( ph -> ps ) )
13 12 imim2i 
 |-  ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ph -> ps ) ) )
14 13 com23 
 |-  ( ( x = A -> ( ph <-> ps ) ) -> ( ph -> ( x = A -> ps ) ) )
15 14 imp 
 |-  ( ( ( x = A -> ( ph <-> ps ) ) /\ ph ) -> ( x = A -> ps ) )
16 15 alanimi 
 |-  ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) -> A. x ( x = A -> ps ) )
17 19.23t 
 |-  ( F/ x ps -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) )
18 17 adantl 
 |-  ( ( F/_ x A /\ F/ x ps ) -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) )
19 16 18 syl5ib 
 |-  ( ( F/_ x A /\ F/ x ps ) -> ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) -> ( E. x x = A -> ps ) ) )
20 19 imp 
 |-  ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) ) -> ( E. x x = A -> ps ) )
21 20 3adant3 
 |-  ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> ( E. x x = A -> ps ) )
22 11 21 mpd 
 |-  ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> ps )