Theorem nfbrd 5188

Description: Deduction version of bound-variable hypothesis builder nfbr 5189. (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 5143	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		nfbrd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfbrd.4	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵)
4	2, 3	nfopd 4889	. . 3 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩)
5		nfbrd.3	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝑅)
6	4, 5	nfeld 2916	. 2 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩ ∈ 𝑅)
7	1, 6	nfxfrd 1853	1 ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)

Syntax hints:   → wi 4  Ⅎwnf 1782   ∈ wcel 2107  Ⅎwnfc 2889  ⟨cop 4631   class class class wbr 5142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143

This theorem is referenced by:  nfbr  5189  nfttrcld  9751