ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfeqd Unicode version

Theorem nfeqd 2208
Description: Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfeqd.1  |-  ( ph  -> 
F/_ x A )
nfeqd.2  |-  ( ph  -> 
F/_ x B )
Assertion
Ref Expression
nfeqd  |-  ( ph  ->  F/ x  A  =  B )

Proof of Theorem nfeqd
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2050 . 2  |-  ( A  =  B  <->  A. y
( y  e.  A  <->  y  e.  B ) )
2 nfv 1437 . . 3  |-  F/ y
ph
3 nfeqd.1 . . . . 5  |-  ( ph  -> 
F/_ x A )
43nfcrd 2207 . . . 4  |-  ( ph  ->  F/ x  y  e.  A )
5 nfeqd.2 . . . . 5  |-  ( ph  -> 
F/_ x B )
65nfcrd 2207 . . . 4  |-  ( ph  ->  F/ x  y  e.  B )
74, 6nfbid 1496 . . 3  |-  ( ph  ->  F/ x ( y  e.  A  <->  y  e.  B ) )
82, 7nfald 1659 . 2  |-  ( ph  ->  F/ x A. y
( y  e.  A  <->  y  e.  B ) )
91, 8nfxfrd 1380 1  |-  ( ph  ->  F/ x  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   A.wal 1257    = wceq 1259   F/wnf 1365    e. wcel 1409   F/_wnfc 2181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-4 1416  ax-17 1435  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-cleq 2049  df-nfc 2183
This theorem is referenced by:  nfeld  2209  nfned  2313  vtoclgft  2621  sbcralt  2862  sbcrext  2863  csbiebt  2914  dfnfc2  3626  eusvnfb  4214  eusv2i  4215  iota2df  4919  riota5f  5520
  Copyright terms: Public domain W3C validator