Theorem vtoclgftOLD 3255

Description: Obsolete proof of vtoclgft 3254 as of 11-Aug-2021. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem vtoclgftOLD

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( A e. V -> A e. _V )
2		elisset 3215	. . . . 5 |- ( A e. _V -> E. z z = A )
3	2	3ad2ant3 1084	. . . 4 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> E. z z = A )
4		nfnfc1 2767	. . . . . . 7 |- F/ x F/_ x A
5		nfcvd 2765	. . . . . . . 8 |- ( F/_ x A -> F/_ x z )
6		id 22	. . . . . . . 8 |- ( F/_ x A -> F/_ x A )
7	5, 6	nfeqd 2772	. . . . . . 7 |- ( F/_ x A -> F/ x z = A )
8		eqeq1 2626	. . . . . . . 8 |- ( z = x -> ( z = A <-> x = A ) )
9	8	a1i 11	. . . . . . 7 |- ( F/_ x A -> ( z = x -> ( z = A <-> x = A ) ) )
10	4, 7, 9	cbvexd 2278	. . . . . 6 |- ( F/_ x A -> ( E. z z = A <-> E. x x = A ) )
11	10	ad2antrr 762	. . . . 5 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) ) -> ( E. z z = A <-> E. x x = A ) )
12	11	3adant3 1081	. . . 4 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> ( E. z z = A <-> E. x x = A ) )
13	3, 12	mpbid 222	. . 3 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> E. x x = A )
14		biimp 205	. . . . . . . . 9 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
15	14	imim2i 16	. . . . . . . 8 |- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ph -> ps ) ) )
16	15	com23 86	. . . . . . 7 |- ( ( x = A -> ( ph <-> ps ) ) -> ( ph -> ( x = A -> ps ) ) )
17	16	imp 445	. . . . . 6 |- ( ( ( x = A -> ( ph <-> ps ) ) /\ ph ) -> ( x = A -> ps ) )
18	17	alanimi 1744	. . . . 5 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) -> A. x ( x = A -> ps ) )
19	18	3ad2ant2 1083	. . . 4 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> A. x ( x = A -> ps ) )
20		simp1r 1086	. . . . 5 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> F/ x ps )
21		19.23t 2079	. . . . 5 |- ( F/ x ps -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) )
22	20, 21	syl 17	. . . 4 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) )
23	19, 22	mpbid 222	. . 3 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> ( E. x x = A -> ps ) )
24	13, 23	mpd 15	. 2 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> ps )
25	1, 24	syl3an3 1361	1 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by: (None)