Theorem optoclOLD 5720

Description: Obsolete version of optocl 5719 as of 29-Dec-2025. (Contributed by NM, 5-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
optocl.1	⊢ 𝐷 = (𝐵 × 𝐶)
optocl.2	⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓))
optocl.3	⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑)

Assertion

Ref	Expression
optoclOLD	⊢ (𝐴 ∈ 𝐷 → 𝜓)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝜓,𝑥,𝑦

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

Proof of Theorem optoclOLD

Step	Hyp	Ref	Expression
1		elxp3 5691	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)))
2		opelxp 5661	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
3		optocl.3	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑)
4	2, 3	sylbi 218	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) → 𝜑)
5		optocl.2	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓))
6	4, 5	imbitrid 245	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) → 𝜓))
7	6	imp 407	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) → 𝜓)
8	7	exlimivv 1939	. . 3 ⊢ (∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) → 𝜓)
9	1, 8	sylbi 218	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝜓)
10		optocl.1	. 2 ⊢ 𝐷 = (𝐵 × 𝐶)
11	9, 10	eleq2s 2858	1 ⊢ (𝐴 ∈ 𝐷 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ⟨cop 4568   × cxp 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-opab 5142  df-xp 5631

This theorem is referenced by: (None)