Theorem nfbrd 4090

Description: Deduction version of bound-variable hypothesis builder nfbr 4091. (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

Step	Hyp	Ref	Expression
1		df-br 4046	. 2 |- ( A R B <-> <. A , B >. e. R )
2		nfbrd.2	. . . 4 |- ( ph -> F/_ x A )
3		nfbrd.4	. . . 4 |- ( ph -> F/_ x B )
4	2, 3	nfopd 3836	. . 3 |- ( ph -> F/_ x <. A , B >. )
5		nfbrd.3	. . 3 |- ( ph -> F/_ x R )
6	4, 5	nfeld 2364	. 2 |- ( ph -> F/ x <. A , B >. e. R )
7	1, 6	nfxfrd 1498	1 |- ( ph -> F/ x A R B )

Syntax hints:   -> wi 4   F/wnf 1483   e. wcel 2176   F/_wnfc 2335   <.cop 3636   class class class wbr 4045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046

This theorem is referenced by:  nfbr  4091