Theorem dfnfc2OLD 4455

Description: Obsolete proof of dfnfc2 4454 as of 26-Jul-2021. (Contributed by Mario Carneiro, 14-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dfnfc2OLD	|- ( A. x A e. V -> ( F/_ x A <-> A. y F/ x y = A ) )

Distinct variable groups:   x, y   y, A

Allowed substitution hints:   A( x)   V( x, y)

Proof of Theorem dfnfc2OLD

Step	Hyp	Ref	Expression
1		nfcvd 2765	. . . 4 |- ( F/_ x A -> F/_ x y )
2		id 22	. . . 4 |- ( F/_ x A -> F/_ x A )
3	1, 2	nfeqd 2772	. . 3 |- ( F/_ x A -> F/ x y = A )
4	3	alrimiv 1855	. 2 |- ( F/_ x A -> A. y F/ x y = A )
5		simpr 477	. . . . . 6 |- ( ( A. x A e. V /\ A. y F/ x y = A ) -> A. y F/ x y = A )
6		df-nfc 2753	. . . . . . 7 |- ( F/_ x { A } <-> A. y F/ x y e. { A } )
7		velsn 4193	. . . . . . . . 9 |- ( y e. { A } <-> y = A )
8	7	nfbii 1778	. . . . . . . 8 |- ( F/ x y e. { A } <-> F/ x y = A )
9	8	albii 1747	. . . . . . 7 |- ( A. y F/ x y e. { A } <-> A. y F/ x y = A )
10	6, 9	bitri 264	. . . . . 6 |- ( F/_ x { A } <-> A. y F/ x y = A )
11	5, 10	sylibr 224	. . . . 5 |- ( ( A. x A e. V /\ A. y F/ x y = A ) -> F/_ x { A } )
12	11	nfunid 4443	. . . 4 |- ( ( A. x A e. V /\ A. y F/ x y = A ) -> F/_ x U. { A } )
13		nfa1 2028	. . . . . 6 |- F/ x A. x A e. V
14		nfnf1 2031	. . . . . . 7 |- F/ x F/ x y = A
15	14	nfal 2153	. . . . . 6 |- F/ x A. y F/ x y = A
16	13, 15	nfan 1828	. . . . 5 |- F/ x ( A. x A e. V /\ A. y F/ x y = A )
17		unisng 4452	. . . . . . 7 |- ( A e. V -> U. { A } = A )
18	17	sps 2055	. . . . . 6 |- ( A. x A e. V -> U. { A } = A )
19	18	adantr 481	. . . . 5 |- ( ( A. x A e. V /\ A. y F/ x y = A ) -> U. { A } = A )
20	16, 19	nfceqdf 2760	. . . 4 |- ( ( A. x A e. V /\ A. y F/ x y = A ) -> ( F/_ x U. { A } <-> F/_ x A ) )
21	12, 20	mpbid 222	. . 3 |- ( ( A. x A e. V /\ A. y F/ x y = A ) -> F/_ x A )
22	21	ex 450	. 2 |- ( A. x A e. V -> ( A. y F/ x y = A -> F/_ x A ) )
23	4, 22	impbid2 216	1 |- ( A. x A e. V -> ( F/_ x A <-> A. y F/ x y = A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   {csn 4177   U.cuni 4436

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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by: (None)