Theorem nfbrd 5144

Description: Deduction version of bound-variable hypothesis builder nfbr 5145. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
nfbrd.2	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfbrd.3	⊢ (𝜑 → Ⅎ𝑥𝑅)
nfbrd.4	⊢ (𝜑 → Ⅎ𝑥𝐵)

Assertion

Ref	Expression
nfbrd	⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)

Proof of Theorem nfbrd

Step	Hyp	Ref	Expression
1		df-br 5099	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		nfbrd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfbrd.4	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵)
4	2, 3	nfopd 4846	. . 3 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩)
5		nfbrd.3	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝑅)
6	4, 5	nfeld 2910	. 2 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩ ∈ 𝑅)
7	1, 6	nfxfrd 1855	1 ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)

Syntax hints:   → wi 4  Ⅎwnf 1784   ∈ wcel 2113  Ⅎwnfc 2883  ⟨cop 4586   class class class wbr 5098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099

This theorem is referenced by:  nfbr  5145  nfttrcld  9621  nfchnd  18536