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

Step	Hyp	Ref	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 )
4	2, 3	nfopd 3780	. . 3 |- ( ph -> F/_ x <. A , B >. )
5		nfbrd.3	. . 3 |- ( ph -> F/_ x R )
6	4, 5	nfeld 2328	. 2 |- ( ph -> F/ x <. A , B >. e. R )
7	1, 6	nfxfrd 1468	1 |- ( ph -> F/ x A R B )

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