Theorem eusvnf 5390

Description: Even if 𝑥 is free in 𝐴, it is effectively bound when 𝐴(𝑥) is single-valued. (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 14-Oct-2016.)

Assertion

Ref	Expression
eusvnf	⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eusvnf

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		euex 2571	. 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → ∃𝑦∀𝑥 𝑦 = 𝐴)
2		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑥𝑧
3		nfcsb1v 3918	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴
4	3	nfeq2 2920	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = ⦋𝑧 / 𝑥⦌𝐴
5		csbeq1a 3907	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
6	5	eqeq2d 2743	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝐴 ↔ 𝑦 = ⦋𝑧 / 𝑥⦌𝐴))
7	2, 4, 6	spcgf 3581	. . . . . . 7 ⊢ (𝑧 ∈ V → (∀𝑥 𝑦 = 𝐴 → 𝑦 = ⦋𝑧 / 𝑥⦌𝐴))
8	7	elv 3480	. . . . . 6 ⊢ (∀𝑥 𝑦 = 𝐴 → 𝑦 = ⦋𝑧 / 𝑥⦌𝐴)
9		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑥𝑤
10		nfcsb1v 3918	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐴
11	10	nfeq2 2920	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = ⦋𝑤 / 𝑥⦌𝐴
12		csbeq1a 3907	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → 𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
13	12	eqeq2d 2743	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑦 = 𝐴 ↔ 𝑦 = ⦋𝑤 / 𝑥⦌𝐴))
14	9, 11, 13	spcgf 3581	. . . . . . 7 ⊢ (𝑤 ∈ V → (∀𝑥 𝑦 = 𝐴 → 𝑦 = ⦋𝑤 / 𝑥⦌𝐴))
15	14	elv 3480	. . . . . 6 ⊢ (∀𝑥 𝑦 = 𝐴 → 𝑦 = ⦋𝑤 / 𝑥⦌𝐴)
16	8, 15	eqtr3d 2774	. . . . 5 ⊢ (∀𝑥 𝑦 = 𝐴 → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
17	16	alrimivv 1931	. . . 4 ⊢ (∀𝑥 𝑦 = 𝐴 → ∀𝑧∀𝑤⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
18		sbnfc2 4436	. . . 4 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑧∀𝑤⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
19	17, 18	sylibr 233	. . 3 ⊢ (∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)
20	19	exlimiv 1933	. 2 ⊢ (∃𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)
21	1, 20	syl 17	1 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)

Syntax hints:   → wi 4  ∀wal 1539   = wceq 1541  ∃wex 1781  ∃!weu 2562  Ⅎwnfc 2883  Vcvv 3474  ⦋csb 3893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-nul 4323

This theorem is referenced by:  eusvnfb  5391  eusv2i  5392