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Theorem nfbrd 4168
Description: Deduction version of bound-variable hypothesis builder nfbr 4169. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nfbrd.2  |-  ( ph  -> 
F/_ x A )
nfbrd.3  |-  ( ph  -> 
F/_ x R )
nfbrd.4  |-  ( ph  -> 
F/_ x B )
Assertion
Ref Expression
nfbrd  |-  ( ph  ->  F/ x  A R B )

Proof of Theorem nfbrd
StepHypRef Expression
1 df-br 4126 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  R )
2 nfbrd.2 . . . 4  |-  ( ph  -> 
F/_ x A )
3 nfbrd.4 . . . 4  |-  ( ph  -> 
F/_ x B )
42, 3nfopd 3915 . . 3  |-  ( ph  -> 
F/_ x <. A ,  B >. )
5 nfbrd.3 . . 3  |-  ( ph  -> 
F/_ x R )
64, 5nfeld 2517 . 2  |-  ( ph  ->  F/ x <. A ,  B >.  e.  R )
71, 6nfxfrd 1576 1  |-  ( ph  ->  F/ x  A R B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1549    e. wcel 1715   F/_wnfc 2489   <.cop 3732   class class class wbr 4125
This theorem is referenced by:  nfbr  4169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126
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