Theorem nfbrd 5124

Description: Deduction version of bound-variable hypothesis builder nfbr 5125. (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 5079	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		nfbrd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfbrd.4	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵)
4	2, 3	nfopd 4826	. . 3 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩)
5		nfbrd.3	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝑅)
6	4, 5	nfeld 2919	. 2 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩ ∈ 𝑅)
7	1, 6	nfxfrd 1859	1 ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)

Syntax hints:   → wi 4  Ⅎwnf 1789   ∈ wcel 2109  Ⅎwnfc 2888  ⟨cop 4572   class class class wbr 5078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079

This theorem is referenced by:  nfbr  5125  nfttrcld  9429