Theorem nfbr 4135

Description: Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
nfbr.1	⊢ Ⅎ𝑥𝐴
nfbr.2	⊢ Ⅎ𝑥𝑅
nfbr.3	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfbr	⊢ Ⅎ𝑥 𝐴𝑅𝐵

Proof of Theorem nfbr

Step	Hyp	Ref	Expression
1		nfbr.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2	1	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
3		nfbr.2	. . . 4 ⊢ Ⅎ𝑥𝑅
4	3	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝑅)
5		nfbr.3	. . . 4 ⊢ Ⅎ𝑥𝐵
6	5	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐵)
7	2, 4, 6	nfbrd 4134	. 2 ⊢ (⊤ → Ⅎ𝑥 𝐴𝑅𝐵)
8	7	mptru 1406	1 ⊢ Ⅎ𝑥 𝐴𝑅𝐵

Syntax hints:  ⊤wtru 1398  Ⅎwnf 1508  Ⅎwnfc 2361   class class class wbr 4088

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089

This theorem is referenced by:  sbcbrg  4143  nfpo  4398  nfso  4399  pofun  4409  nfse  4438  nffrfor  4445  nfwe  4452  nfco  4895  nfcnv  4909  dfdmf  4924  dfrnf  4973  nfdm  4976  dffun6f  5339  dffun4f  5342  nffv  5649  funfvdm2f  5711  fvmptss2  5721  f1ompt  5798  fmptco  5813  nfiso  5946  nfofr  6241  ofrfval2  6251  tposoprab  6445  modom  6993  xpcomco  7009  nfsup  7190  caucvgprprlemaddq  7927  lble  9126  nfsum1  11916  nfsum  11917  fsum00  12022  mertenslem2  12096  nfcprod1  12114  nfcprod  12115  fprodap0  12181  fprodrec  12189  fproddivapf  12191  fprodap0f  12196  fprodle  12200  oddpwdclemdvds  12741  oddpwdclemndvds  12742