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Theorem nfbrd 4215
Description: Deduction version of bound-variable hypothesis builder nfbr 4216. (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 4173 . 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 3961 . . 3  |-  ( ph  -> 
F/_ x <. A ,  B >. )
5 nfbrd.3 . . 3  |-  ( ph  -> 
F/_ x R )
64, 5nfeld 2555 . 2  |-  ( ph  ->  F/ x <. A ,  B >.  e.  R )
71, 6nfxfrd 1577 1  |-  ( ph  ->  F/ x  A R B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1550    e. wcel 1721   F/_wnfc 2527   <.cop 3777   class class class wbr 4172
This theorem is referenced by:  nfbr  4216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173
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