Theorem opelopab 5548

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.)

Hypotheses

Ref	Expression
opelopab.1	⊢ 𝐴 ∈ V
opelopab.2	⊢ 𝐵 ∈ V
opelopab.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
opelopab.4	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
opelopab	⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜒,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem opelopab

Step	Hyp	Ref	Expression
1		opelopab.1	. 2 ⊢ 𝐴 ∈ V
2		opelopab.2	. 2 ⊢ 𝐵 ∈ V
3		opelopab.3	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4		opelopab.4	. . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
5	3, 4	opelopabg 5544	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
6	1, 2, 5	mp2an 690	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1534   ∈ wcel 2099  Vcvv 3462  ⟨cop 4639  {copab 5215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-opab 5216

This theorem is referenced by:  opabid2  5834  dfres2  6050  f1oiso  7363  elopabi  8076  xporderlem  8141  cnlnssadj  32013  areacirclem5  37413  dicopelval  40876  dih1dimatlem  41028  pellexlem3  42488  fsovrfovd  43676