Theorem eusvnf 4861

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	|- ( E! y A. x y = A -> F/_ x A )

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 2494	. 2 |- ( E! y A. x y = A -> E. y A. x y = A )
2		vex 3203	. . . . . . 7 |- z e. _V
3		nfcv 2764	. . . . . . . 8 |- F/_ x z
4		nfcsb1v 3549	. . . . . . . . 9 |- F/_ x [_ z / x ]_ A
5	4	nfeq2 2780	. . . . . . . 8 |- F/ x y = [_ z / x ]_ A
6		csbeq1a 3542	. . . . . . . . 9 |- ( x = z -> A = [_ z / x ]_ A )
7	6	eqeq2d 2632	. . . . . . . 8 |- ( x = z -> ( y = A <-> y = [_ z / x ]_ A ) )
8	3, 5, 7	spcgf 3288	. . . . . . 7 |- ( z e. _V -> ( A. x y = A -> y = [_ z / x ]_ A ) )
9	2, 8	ax-mp 5	. . . . . 6 |- ( A. x y = A -> y = [_ z / x ]_ A )
10		vex 3203	. . . . . . 7 |- w e. _V
11		nfcv 2764	. . . . . . . 8 |- F/_ x w
12		nfcsb1v 3549	. . . . . . . . 9 |- F/_ x [_ w / x ]_ A
13	12	nfeq2 2780	. . . . . . . 8 |- F/ x y = [_ w / x ]_ A
14		csbeq1a 3542	. . . . . . . . 9 |- ( x = w -> A = [_ w / x ]_ A )
15	14	eqeq2d 2632	. . . . . . . 8 |- ( x = w -> ( y = A <-> y = [_ w / x ]_ A ) )
16	11, 13, 15	spcgf 3288	. . . . . . 7 |- ( w e. _V -> ( A. x y = A -> y = [_ w / x ]_ A ) )
17	10, 16	ax-mp 5	. . . . . 6 |- ( A. x y = A -> y = [_ w / x ]_ A )
18	9, 17	eqtr3d 2658	. . . . 5 |- ( A. x y = A -> [_ z / x ]_ A = [_ w / x ]_ A )
19	18	alrimivv 1856	. . . 4 |- ( A. x y = A -> A. z A. w [_ z / x ]_ A = [_ w / x ]_ A )
20		sbnfc2 4007	. . . 4 |- ( F/_ x A <-> A. z A. w [_ z / x ]_ A = [_ w / x ]_ A )
21	19, 20	sylibr 224	. . 3 |- ( A. x y = A -> F/_ x A )
22	21	exlimiv 1858	. 2 |- ( E. y A. x y = A -> F/_ x A )
23	1, 22	syl 17	1 |- ( E! y A. x y = A -> F/_ x A )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   F/_wnfc 2751   _Vcvv 3200   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916

This theorem is referenced by:  eusvnfb  4862  eusv2i  4863