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Theorem nfned 2784
Description: Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfned.1  |-  ( ph  -> 
F/_ x A )
nfned.2  |-  ( ph  -> 
F/_ x B )
Assertion
Ref Expression
nfned  |-  ( ph  ->  F/ x  A  =/= 
B )

Proof of Theorem nfned
StepHypRef Expression
1 df-ne 2650 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
2 nfned.1 . . . 4  |-  ( ph  -> 
F/_ x A )
3 nfned.2 . . . 4  |-  ( ph  -> 
F/_ x B )
42, 3nfeqd 2623 . . 3  |-  ( ph  ->  F/ x  A  =  B )
54nfnd 1841 . 2  |-  ( ph  ->  F/ x  -.  A  =  B )
61, 5nfxfrd 1617 1  |-  ( ph  ->  F/ x  A  =/= 
B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1370   F/wnf 1590   F/_wnfc 2602    =/= wne 2648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-cleq 2446  df-nfc 2604  df-ne 2650
This theorem is referenced by: (None)
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