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	⊢ (∀x A ∈ V → (ℲxA ↔ ∀yℲx 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 ⊢ (ℲxA → Ⅎxy)
2		id 19	. . . 4 ⊢ (ℲxA → ℲxA)
3	1, 2	nfeqd 2504	. . 3 ⊢ (ℲxA → Ⅎx y = A)
4	3	alrimiv 1631	. 2 ⊢ (ℲxA → ∀yℲx y = A)
5		simpr 447	. . . . . 6 ⊢ ((∀x A ∈ V ∧ ∀yℲx y = A) → ∀yℲx y = A)
6		df-nfc 2479	. . . . . . 7 ⊢ (Ⅎx{A} ↔ ∀yℲx y ∈ {A})
7		elsn 3749	. . . . . . . . 9 ⊢ (y ∈ {A} ↔ y = A)
8	7	nfbii 1569	. . . . . . . 8 ⊢ (Ⅎx y ∈ {A} ↔ Ⅎx y = A)
9	8	albii 1566	. . . . . . 7 ⊢ (∀yℲx y ∈ {A} ↔ ∀yℲx y = A)
10	6, 9	bitri 240	. . . . . 6 ⊢ (Ⅎx{A} ↔ ∀yℲx y = A)
11	5, 10	sylibr 203	. . . . 5 ⊢ ((∀x A ∈ V ∧ ∀yℲx y = A) → Ⅎx{A})
12	11	nfunid 3899	. . . 4 ⊢ ((∀x A ∈ V ∧ ∀yℲx y = A) → Ⅎx∪{A})
13		nfa1 1788	. . . . . 6 ⊢ Ⅎx∀x A ∈ V
14		nfnf1 1790	. . . . . . 7 ⊢ ℲxℲx y = A
15	14	nfal 1842	. . . . . 6 ⊢ Ⅎx∀yℲx y = A
16	13, 15	nfan 1824	. . . . 5 ⊢ Ⅎx(∀x A ∈ V ∧ ∀yℲx y = A)
17		unisng 3909	. . . . . . 7 ⊢ (A ∈ V → ∪{A} = A)
18	17	sps 1754	. . . . . 6 ⊢ (∀x A ∈ V → ∪{A} = A)
19	18	adantr 451	. . . . 5 ⊢ ((∀x A ∈ V ∧ ∀yℲx y = A) → ∪{A} = A)
20	16, 19	nfceqdf 2489	. . . 4 ⊢ ((∀x A ∈ V ∧ ∀yℲx y = A) → (Ⅎx∪{A} ↔ ℲxA))
21	12, 20	mpbid 201	. . 3 ⊢ ((∀x A ∈ V ∧ ∀yℲx y = A) → ℲxA)
22	21	ex 423	. 2 ⊢ (∀x A ∈ V → (∀yℲx y = A → ℲxA))
23	4, 22	impbid2 195	1 ⊢ (∀x A ∈ V → (ℲxA ↔ ∀yℲx y = A))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  {csn 3738  ∪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)