Metamath Proof Explorer

Theorem vtoclgft

Description: Closed theorem form of vtoclgf . (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 Gino Giotto, 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 elisset 
 |-  ( A e. V -> E. z z = A )
2 nfv 
 |-  F/ z F/_ x A
3 nfnfc1 
 |-  F/ x F/_ x A
4 nfcvd 
 |-  ( F/_ x A -> F/_ x z )
5 id 
 |-  ( F/_ x A -> F/_ x A )
6 4 5 nfeqd 
 |-  ( F/_ x A -> F/ x z = A )
7 6 nfnd 
 |-  ( F/_ x A -> F/ x -. z = A )
8 nfvd 
 |-  ( F/_ x A -> F/ z -. x = A )
9 eqeq1 
 |-  ( z = x -> ( z = A <-> x = A ) )
10 9 a1i 
 |-  ( F/_ x A -> ( z = x -> ( z = A <-> x = A ) ) )
11 notbi 
 |-  ( ( z = A <-> x = A ) <-> ( -. z = A <-> -. x = A ) )
12 10 11 syl6ib 
 |-  ( F/_ x A -> ( z = x -> ( -. z = A <-> -. x = A ) ) )
13 biimp 
 |-  ( ( -. z = A <-> -. x = A ) -> ( -. z = A -> -. x = A ) )
14 12 13 syl6 
 |-  ( F/_ x A -> ( z = x -> ( -. z = A -> -. x = A ) ) )
15 2 3 7 8 14 cbv1v 
 |-  ( F/_ x A -> ( A. z -. z = A -> A. x -. x = A ) )
16 equcomi 
 |-  ( x = z -> z = x )
17 biimpr 
 |-  ( ( -. z = A <-> -. x = A ) -> ( -. x = A -> -. z = A ) )
18 16 12 17 syl56 
 |-  ( F/_ x A -> ( x = z -> ( -. x = A -> -. z = A ) ) )
19 3 2 8 7 18 cbv1v 
 |-  ( F/_ x A -> ( A. x -. x = A -> A. z -. z = A ) )
20 15 19 impbid 
 |-  ( F/_ x A -> ( A. z -. z = A <-> A. x -. x = A ) )
21 alnex 
 |-  ( A. z -. z = A <-> -. E. z z = A )
22 alnex 
 |-  ( A. x -. x = A <-> -. E. x x = A )
23 20 21 22 3bitr3g 
 |-  ( F/_ x A -> ( -. E. z z = A <-> -. E. x x = A ) )
24 23 con4bid 
 |-  ( F/_ x A -> ( E. z z = A <-> E. x x = A ) )
25 1 24 syl5ib 
 |-  ( F/_ x A -> ( A e. V -> E. x x = A ) )
26 25 ad2antrr 
 |-  ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) ) -> ( A e. V -> E. x x = A ) )
27 26 3impia 
 |-  ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> E. x x = A )
28 biimp 
 |-  ( ( ph <-> ps ) -> ( ph -> ps ) )
29 28 imim2i 
 |-  ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ph -> ps ) ) )
30 29 com23 
 |-  ( ( x = A -> ( ph <-> ps ) ) -> ( ph -> ( x = A -> ps ) ) )
31 30 imp 
 |-  ( ( ( x = A -> ( ph <-> ps ) ) /\ ph ) -> ( x = A -> ps ) )
32 31 alanimi 
 |-  ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) -> A. x ( x = A -> ps ) )
33 19.23t 
 |-  ( F/ x ps -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) )
34 33 adantl 
 |-  ( ( F/_ x A /\ F/ x ps ) -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) )
35 32 34 syl5ib 
 |-  ( ( F/_ x A /\ F/ x ps ) -> ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) -> ( E. x x = A -> ps ) ) )
36 35 imp 
 |-  ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) ) -> ( E. x x = A -> ps ) )
37 36 3adant3 
 |-  ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> ( E. x x = A -> ps ) )
38 27 37 mpd 
 |-  ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> ps )