Theorem nfbrd 4074

Description: Deduction version of bound-variable hypothesis builder nfbr 4075. (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 4030	. 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 3821	. . 3 |- ( ph -> F/_ x <. A , B >. )
5		nfbrd.3	. . 3 |- ( ph -> F/_ x R )
6	4, 5	nfeld 2352	. 2 |- ( ph -> F/ x <. A , B >. e. R )
7	1, 6	nfxfrd 1486	1 |- ( ph -> F/ x A R B )

Syntax hints:   -> wi 4   F/wnf 1471   e. wcel 2164   F/_wnfc 2323   <.cop 3621   class class class wbr 4029

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030

This theorem is referenced by:  nfbr  4075