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Theorem nfned 2690
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 2600 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
2 nfned.1 . . . 4  |-  ( ph  -> 
F/_ x A )
3 nfned.2 . . . 4  |-  ( ph  -> 
F/_ x B )
42, 3nfeqd 2585 . . 3  |-  ( ph  ->  F/ x  A  =  B )
54nfnd 1809 . 2  |-  ( ph  ->  F/ x  -.  A  =  B )
61, 5nfxfrd 1580 1  |-  ( ph  ->  F/ x  A  =/= 
B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   F/wnf 1553    = wceq 1652   F/_wnfc 2558    =/= wne 2598
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-cleq 2428  df-nfc 2560  df-ne 2600
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