Theorem opelopabt 5537

Description: Closed theorem form of opelopab 5547. (Contributed by NM, 19-Feb-2013.)

Assertion

Ref	Expression
opelopabt	⊢ ((∀𝑥∀𝑦(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥∀𝑦(𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (⟨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)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem opelopabt

Step	Hyp	Ref	Expression
1		elopab 5532	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2		19.26-2 1871	. . . 4 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))) ↔ (∀𝑥∀𝑦(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥∀𝑦(𝑦 = 𝐵 → (𝜓 ↔ 𝜒))))
3		anim12 809	. . . . . 6 ⊢ (((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))) → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝜑 ↔ 𝜓) ∧ (𝜓 ↔ 𝜒))))
4		bitr 805	. . . . . 6 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜓 ↔ 𝜒)) → (𝜑 ↔ 𝜒))
5	3, 4	syl6 35	. . . . 5 ⊢ (((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))) → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜒)))
6	5	2alimi 1812	. . . 4 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))) → ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜒)))
7	2, 6	sylbir 235	. . 3 ⊢ ((∀𝑥∀𝑦(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥∀𝑦(𝑦 = 𝐵 → (𝜓 ↔ 𝜒))) → ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜒)))
8		copsex2t 5497	. . 3 ⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜒)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜒))
9	7, 8	stoic3 1776	. 2 ⊢ ((∀𝑥∀𝑦(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥∀𝑦(𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜒))
10	1, 9	bitrid 283	1 ⊢ ((∀𝑥∀𝑦(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥∀𝑦(𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ⟨cop 4632  {copab 5205

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-opab 5206

This theorem is referenced by: (None)