MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eusvnf Structured version   Unicode version

Theorem eusvnf 4487
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.
StepHypRef Expression
1 euex 2281 . 2  |-  ( E! y A. x  y  =  A  ->  E. y A. x  y  =  A )
2 vex 2975 . . . . . . 7  |-  z  e. 
_V
3 nfcv 2579 . . . . . . . 8  |-  F/_ x
z
4 nfcsb1v 3304 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ A
54nfeq2 2590 . . . . . . . 8  |-  F/ x  y  =  [_ z  /  x ]_ A
6 csbeq1a 3297 . . . . . . . . 9  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
76eqeq2d 2454 . . . . . . . 8  |-  ( x  =  z  ->  (
y  =  A  <->  y  =  [_ z  /  x ]_ A ) )
83, 5, 7spcgf 3052 . . . . . . 7  |-  ( z  e.  _V  ->  ( A. x  y  =  A  ->  y  =  [_ z  /  x ]_ A
) )
92, 8ax-mp 5 . . . . . 6  |-  ( A. x  y  =  A  ->  y  =  [_ z  /  x ]_ A )
10 vex 2975 . . . . . . 7  |-  w  e. 
_V
11 nfcv 2579 . . . . . . . 8  |-  F/_ x w
12 nfcsb1v 3304 . . . . . . . . 9  |-  F/_ x [_ w  /  x ]_ A
1312nfeq2 2590 . . . . . . . 8  |-  F/ x  y  =  [_ w  /  x ]_ A
14 csbeq1a 3297 . . . . . . . . 9  |-  ( x  =  w  ->  A  =  [_ w  /  x ]_ A )
1514eqeq2d 2454 . . . . . . . 8  |-  ( x  =  w  ->  (
y  =  A  <->  y  =  [_ w  /  x ]_ A ) )
1611, 13, 15spcgf 3052 . . . . . . 7  |-  ( w  e.  _V  ->  ( A. x  y  =  A  ->  y  =  [_ w  /  x ]_ A
) )
1710, 16ax-mp 5 . . . . . 6  |-  ( A. x  y  =  A  ->  y  =  [_ w  /  x ]_ A )
189, 17eqtr3d 2477 . . . . 5  |-  ( A. x  y  =  A  ->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
1918alrimivv 1686 . . . 4  |-  ( A. x  y  =  A  ->  A. z A. w [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
20 sbnfc2 3706 . . . 4  |-  ( F/_ x A  <->  A. z A. w [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
2119, 20sylibr 212 . . 3  |-  ( A. x  y  =  A  -> 
F/_ x A )
2221exlimiv 1688 . 2  |-  ( E. y A. x  y  =  A  ->  F/_ x A )
231, 22syl 16 1  |-  ( E! y A. x  y  =  A  ->  F/_ x A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756   E!weu 2253   F/_wnfc 2566   _Vcvv 2972   [_csb 3288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-in 3335  df-ss 3342  df-nul 3638
This theorem is referenced by:  eusvnfb  4488  eusv2i  4489
  Copyright terms: Public domain W3C validator