Theorem brabsb 5491

Description: The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.)

Hypothesis

Ref	Expression
brabsb.1	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Assertion

Ref	Expression
brabsb	⊢ (𝐴𝑅𝐵 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)

Distinct variable groups:   𝑥,𝑦   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem brabsb

Step	Hyp	Ref	Expression
1		df-br 5108	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		brabsb.1	. . 3 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
3	2	eleq2i 2820	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4		opelopabsb 5490	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
5	1, 3, 4	3bitri 297	1 ⊢ (𝐴𝑅𝐵 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  [wsbc 3753  ⟨cop 4595   class class class wbr 5107  {copab 5169

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170

This theorem is referenced by:  eqerlem  8706  xpab  35713