Theorem nfbrd 5127

Description: Deduction version of bound-variable hypothesis builder nfbr 5128. (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 5082	. 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 2916	. 2 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩ ∈ 𝑅)
7	1, 6	nfxfrd 1854	1 ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)

Syntax hints:   → wi 4  Ⅎwnf 1783   ∈ wcel 2104  Ⅎwnfc 2885  ⟨cop 4571   class class class wbr 5081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082

This theorem is referenced by:  nfbr  5128  nfttrcld  9512