Theorem dfnfc2 3589

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

Assertion

Ref	Expression
dfnfc2	|- (A.x A e. V -> ( F/_xA <-> 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 2176	. . . 4 |- ( F/_xA -> F/_xy)
2		id 19	. . . 4 |- ( F/_xA -> F/_xA)
3	1, 2	nfeqd 2189	. . 3 |- ( F/_xA -> F/x y = A)
4	3	alrimiv 1751	. 2 |- ( F/_xA -> A.y F/x y = A)
5		simpr 103	. . . . . 6 |- ((A.x A e. V /\ A.y F/x y = A) -> A.y F/x y = A)
6		df-nfc 2164	. . . . . . 7 |- ( F/_x {A } <-> A.y F/x y e. {A })
7		elsn 3382	. . . . . . . . 9 |- (y e. {A } <-> y = A)
8	7	nfbii 1359	. . . . . . . 8 |- ( F/x y e. {A } <-> F/x y = A)
9	8	albii 1356	. . . . . . 7 |- (A.y F/x y e. {A } <-> A.y F/x y = A)
10	6, 9	bitri 173	. . . . . 6 |- ( F/_x {A } <-> A.y F/x y = A)
11	5, 10	sylibr 137	. . . . 5 |- ((A.x A e. V /\ A.y F/x y = A) -> F/_x {A })
12	11	nfunid 3578	. . . 4 |- ((A.x A e. V /\ A.y F/x y = A) -> F/_x U. {A })
13		nfa1 1431	. . . . . 6 |- F/xA.x A e. V
14		nfnf1 1433	. . . . . . 7 |- F/x F/x y = A
15	14	nfal 1465	. . . . . 6 |- F/xA.y F/x y = A
16	13, 15	nfan 1454	. . . . 5 |- F/x(A.x A e. V /\ A.y F/x y = A)
17		unisng 3588	. . . . . . 7 |- (A e. V -> U. {A } = A)
18	17	sps 1427	. . . . . 6 |- (A.x A e. V -> U. {A } = A)
19	18	adantr 261	. . . . 5 |- ((A.x A e. V /\ A.y F/x y = A) -> U. {A } = A)
20	16, 19	nfceqdf 2174	. . . 4 |- ((A.x A e. V /\ A.y F/x y = A) -> ( F/_x U. {A } <-> F/_xA))
21	12, 20	mpbid 135	. . 3 |- ((A.x A e. V /\ A.y F/x y = A) -> F/_xA)
22	21	ex 108	. 2 |- (A.x A e. V -> (A.y F/x y = A -> F/_xA))
23	4, 22	impbid2 131	1 |- (A.x A e. V -> ( F/_xA <-> A.y F/x y = A))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240   = wceq 1242   F/wnf 1346   e. wcel 1390   F/_wnfc 2162   {csn 3367   U.cuni 3571

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572

This theorem is referenced by:  eusv2nf  4154