Theorem dfnfc2 3910

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 2491	. . . 4 |- ( F/_xA -> F/_xy)
2		id 19	. . . 4 |- ( F/_xA -> F/_xA)
3	1, 2	nfeqd 2504	. . 3 |- ( F/_xA -> F/x y = A)
4	3	alrimiv 1631	. 2 |- ( F/_xA -> A.y F/x y = A)
5		simpr 447	. . . . . 6 |- ((A.x A e. V /\ A.y F/x y = A) -> A.y F/x y = A)
6		df-nfc 2479	. . . . . . 7 |- ( F/_x{A} <-> A.y F/x y e. {A})
7		elsn 3749	. . . . . . . . 9 |- (y e. {A} <-> y = A)
8	7	nfbii 1569	. . . . . . . 8 |- ( F/x y e. {A} <-> F/x y = A)
9	8	albii 1566	. . . . . . 7 |- (A.y F/x y e. {A} <-> A.y F/x y = A)
10	6, 9	bitri 240	. . . . . 6 |- ( F/_x{A} <-> A.y F/x y = A)
11	5, 10	sylibr 203	. . . . 5 |- ((A.x A e. V /\ A.y F/x y = A) -> F/_x{A})
12	11	nfunid 3899	. . . 4 |- ((A.x A e. V /\ A.y F/x y = A) -> F/_xU.{A})
13		nfa1 1788	. . . . . 6 |- F/xA.x A e. V
14		nfnf1 1790	. . . . . . 7 |- F/x F/x y = A
15	14	nfal 1842	. . . . . 6 |- F/xA.y F/x y = A
16	13, 15	nfan 1824	. . . . 5 |- F/x(A.x A e. V /\ A.y F/x y = A)
17		unisng 3909	. . . . . . 7 |- (A e. V -> U.{A} = A)
18	17	sps 1754	. . . . . 6 |- (A.x A e. V -> U.{A} = A)
19	18	adantr 451	. . . . 5 |- ((A.x A e. V /\ A.y F/x y = A) -> U.{A} = A)
20	16, 19	nfceqdf 2489	. . . 4 |- ((A.x A e. V /\ A.y F/x y = A) -> ( F/_xU.{A} <-> F/_xA))
21	12, 20	mpbid 201	. . 3 |- ((A.x A e. V /\ A.y F/x y = A) -> F/_xA)
22	21	ex 423	. 2 |- (A.x A e. V -> (A.y F/x y = A -> F/_xA))
23	4, 22	impbid2 195	1 |- (A.x A e. V -> ( F/_xA <-> A.y F/x y = A))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   F/wnf 1544   = wceq 1642   e. wcel 1710   F/_wnfc 2477  {csn 3738  U.cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-uni 3893

This theorem is referenced by: (None)