Theorem eusvnf 4447

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 2054	. 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → ∃𝑦∀𝑥 𝑦 = 𝐴)
2		vex 2738	. . . . . . 7 ⊢ 𝑧 ∈ V
3		nfcv 2317	. . . . . . . 8 ⊢ Ⅎ𝑥𝑧
4		nfcsb1v 3088	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴
5	4	nfeq2 2329	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = ⦋𝑧 / 𝑥⦌𝐴
6		csbeq1a 3064	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
7	6	eqeq2d 2187	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝐴 ↔ 𝑦 = ⦋𝑧 / 𝑥⦌𝐴))
8	3, 5, 7	spcgf 2817	. . . . . . 7 ⊢ (𝑧 ∈ V → (∀𝑥 𝑦 = 𝐴 → 𝑦 = ⦋𝑧 / 𝑥⦌𝐴))
9	2, 8	ax-mp 5	. . . . . 6 ⊢ (∀𝑥 𝑦 = 𝐴 → 𝑦 = ⦋𝑧 / 𝑥⦌𝐴)
10		vex 2738	. . . . . . 7 ⊢ 𝑤 ∈ V
11		nfcv 2317	. . . . . . . 8 ⊢ Ⅎ𝑥𝑤
12		nfcsb1v 3088	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐴
13	12	nfeq2 2329	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = ⦋𝑤 / 𝑥⦌𝐴
14		csbeq1a 3064	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → 𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
15	14	eqeq2d 2187	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑦 = 𝐴 ↔ 𝑦 = ⦋𝑤 / 𝑥⦌𝐴))
16	11, 13, 15	spcgf 2817	. . . . . . 7 ⊢ (𝑤 ∈ V → (∀𝑥 𝑦 = 𝐴 → 𝑦 = ⦋𝑤 / 𝑥⦌𝐴))
17	10, 16	ax-mp 5	. . . . . 6 ⊢ (∀𝑥 𝑦 = 𝐴 → 𝑦 = ⦋𝑤 / 𝑥⦌𝐴)
18	9, 17	eqtr3d 2210	. . . . 5 ⊢ (∀𝑥 𝑦 = 𝐴 → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
19	18	alrimivv 1873	. . . 4 ⊢ (∀𝑥 𝑦 = 𝐴 → ∀𝑧∀𝑤⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
20		sbnfc2 3115	. . . 4 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑧∀𝑤⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
21	19, 20	sylibr 134	. . 3 ⊢ (∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)
22	21	exlimiv 1596	. 2 ⊢ (∃𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)
23	1, 22	syl 14	1 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)

Syntax hints:   → wi 4  ∀wal 1351   = wceq 1353  ∃wex 1490  ∃!weu 2024   ∈ wcel 2146  Ⅎwnfc 2304  Vcvv 2735  ⦋csb 3055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-sbc 2961  df-csb 3056

This theorem is referenced by:  eusvnfb  4448  eusv2i  4449