Theorem nfbrd 3807

Description: Deduction version of bound-variable hypothesis builder nfbr 3808. (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 3765	. 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 3566	. . 3 |- ( ph -> F/_ x <. A , B >. )
5		nfbrd.3	. . 3 |- ( ph -> F/_ x R )
6	4, 5	nfeld 2193	. 2 |- ( ph -> F/ x <. A , B >. e. R )
7	1, 6	nfxfrd 1364	1 |- ( ph -> F/ x A R B )

Syntax hints:   -> wi 4   F/wnf 1349   e. wcel 1393   F/_wnfc 2165   <.cop 3378   class class class wbr 3764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765

This theorem is referenced by:  nfbr  3808