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Theorem nfbrd 4032
Description: Deduction version of bound-variable hypothesis builder nfbr 4033. (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 3988 . 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 3780 . . 3  |-  ( ph  -> 
F/_ x <. A ,  B >. )
5 nfbrd.3 . . 3  |-  ( ph  -> 
F/_ x R )
64, 5nfeld 2328 . 2  |-  ( ph  ->  F/ x <. A ,  B >.  e.  R )
71, 6nfxfrd 1468 1  |-  ( ph  ->  F/ x  A R B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1453    e. wcel 2141   F/_wnfc 2299   <.cop 3584   class class class wbr 3987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988
This theorem is referenced by:  nfbr  4033
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