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Theorem nfbrd 4445
Description: Deduction version of bound-variable hypothesis builder nfbr 4446. (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 4402 . 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 4182 . . 3  |-  ( ph  -> 
F/_ x <. A ,  B >. )
5 nfbrd.3 . . 3  |-  ( ph  -> 
F/_ x R )
64, 5nfeld 2599 . 2  |-  ( ph  ->  F/ x <. A ,  B >.  e.  R )
71, 6nfxfrd 1696 1  |-  ( ph  ->  F/ x  A R B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   F/wnf 1666    e. wcel 1886   F/_wnfc 2578   <.cop 3973   class class class wbr 4401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-rab 2745  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-br 4402
This theorem is referenced by:  nfbr  4446
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