Theorem nfbrd 5132

Description: Deduction version of bound-variable hypothesis builder nfbr 5133. (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 5087	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		nfbrd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfbrd.4	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵)
4	2, 3	nfopd 4834	. . 3 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩)
5		nfbrd.3	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝑅)
6	4, 5	nfeld 2911	. 2 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩ ∈ 𝑅)
7	1, 6	nfxfrd 1856	1 ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)

Syntax hints:   → wi 4  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2884  ⟨cop 4574   class class class wbr 5086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087

This theorem is referenced by:  nfbr  5133  nfttrcld  9626  nfchnd  18572