Theorem dfnfc2 4454

Description: An alternative statement of the effective freeness of a class A, when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.) (Proof shortened by JJ, 26-Jul-2021.)

Assertion

Ref	Expression
dfnfc2	|- ( 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 dfnfc2

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		df-nfc 2753	. . . . 5 |- ( F/_ x { A } <-> A. y F/ x y e. { A } )
6		velsn 4193	. . . . . . 7 |- ( y e. { A } <-> y = A )
7	6	nfbii 1778	. . . . . 6 |- ( F/ x y e. { A } <-> F/ x y = A )
8	7	albii 1747	. . . . 5 |- ( A. y F/ x y e. { A } <-> A. y F/ x y = A )
9	5, 8	sylbbr 226	. . . 4 |- ( A. y F/ x y = A -> F/_ x { A } )
10	9	nfunid 4443	. . 3 |- ( A. y F/ x y = A -> F/_ x U. { A } )
11		nfa1 2028	. . . 4 |- F/ x A. x A e. V
12		unisng 4452	. . . . 5 |- ( A e. V -> U. { A } = A )
13	12	sps 2055	. . . 4 |- ( A. x A e. V -> U. { A } = A )
14	11, 13	nfceqdf 2760	. . 3 |- ( A. x A e. V -> ( F/_ x U. { A } <-> F/_ x A ) )
15	10, 14	syl5ib 234	. 2 |- ( A. x A e. V -> ( A. y F/ x y = A -> F/_ x A ) )
16	4, 15	impbid2 216	1 |- ( A. x A e. V -> ( F/_ x A <-> A. y F/ x y = A ) )

Syntax hints:   -> wi 4   <-> wb 196   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:  eusv2nf  4864