Theorem eusvnf 4131

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 1908	. 2 ⊢ (∃!y∀x y = A → ∃y∀x y = A)
2		vex 2534	. . . . . . 7 ⊢ z ∈ V
3		nfcv 2156	. . . . . . . 8 ⊢ Ⅎxz
4		nfcsb1v 2855	. . . . . . . . 9 ⊢ Ⅎx⦋z / x⦌A
5	4	nfeq2 2167	. . . . . . . 8 ⊢ Ⅎx y = ⦋z / x⦌A
6		csbeq1a 2833	. . . . . . . . 9 ⊢ (x = z → A = ⦋z / x⦌A)
7	6	eqeq2d 2029	. . . . . . . 8 ⊢ (x = z → (y = A ↔ y = ⦋z / x⦌A))
8	3, 5, 7	spcgf 2608	. . . . . . 7 ⊢ (z ∈ V → (∀x y = A → y = ⦋z / x⦌A))
9	2, 8	ax-mp 7	. . . . . 6 ⊢ (∀x y = A → y = ⦋z / x⦌A)
10		vex 2534	. . . . . . 7 ⊢ w ∈ V
11		nfcv 2156	. . . . . . . 8 ⊢ Ⅎxw
12		nfcsb1v 2855	. . . . . . . . 9 ⊢ Ⅎx⦋w / x⦌A
13	12	nfeq2 2167	. . . . . . . 8 ⊢ Ⅎx y = ⦋w / x⦌A
14		csbeq1a 2833	. . . . . . . . 9 ⊢ (x = w → A = ⦋w / x⦌A)
15	14	eqeq2d 2029	. . . . . . . 8 ⊢ (x = w → (y = A ↔ y = ⦋w / x⦌A))
16	11, 13, 15	spcgf 2608	. . . . . . 7 ⊢ (w ∈ V → (∀x y = A → y = ⦋w / x⦌A))
17	10, 16	ax-mp 7	. . . . . 6 ⊢ (∀x y = A → y = ⦋w / x⦌A)
18	9, 17	eqtr3d 2052	. . . . 5 ⊢ (∀x y = A → ⦋z / x⦌A = ⦋w / x⦌A)
19	18	alrimivv 1733	. . . 4 ⊢ (∀x y = A → ∀z∀w⦋z / x⦌A = ⦋w / x⦌A)
20		sbnfc2 2879	. . . 4 ⊢ (ℲxA ↔ ∀z∀w⦋z / x⦌A = ⦋w / x⦌A)
21	19, 20	sylibr 137	. . 3 ⊢ (∀x y = A → ℲxA)
22	21	exlimiv 1467	. 2 ⊢ (∃y∀x y = A → ℲxA)
23	1, 22	syl 14	1 ⊢ (∃!y∀x y = A → ℲxA)

Syntax hints:   → wi 4  ∀wal 1224   = wceq 1226  ∃wex 1358   ∈ wcel 1370  ∃!weu 1878  Ⅎwnfc 2143  Vcvv 2531  ⦋csb 2825

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000

This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-sbc 2738  df-csb 2826

This theorem is referenced by:  eusvnfb  4132  eusv2i  4133