Theorem nfbrd 4155

Description: Deduction version of bound-variable hypothesis builder nfbr 4156. (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 4110	. 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 3900	. . 3 |- ( ph -> F/_ x <. A , B >. )
5		nfbrd.3	. . 3 |- ( ph -> F/_ x R )
6	4, 5	nfeld 2400	. 2 |- ( ph -> F/ x <. A , B >. e. R )
7	1, 6	nfxfrd 1524	1 |- ( ph -> F/ x A R B )

Syntax hints:   -> wi 4   F/wnf 1509   e. wcel 2203   F/_wnfc 2371   <.cop 3692   class class class wbr 4109

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110

This theorem is referenced by:  nfbr  4156