Theorem optocl 5754

Description: Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.) Shorten and reduce axiom usage. (Revised by TM, 29-Dec-2025.)

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
optocl	⊢ (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 optocl

Step	Hyp	Ref	Expression
1		elxpi 5682	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
2		optocl.3	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑)
3		optocl.2	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓))
4	3	eqcoms 2770	. . . . . 6 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))
5	2, 4	imbitrid 247	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜓))
6	5	imp 411	. . . 4 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) → 𝜓)
7	6	exlimivv 1961	. . 3 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) → 𝜓)
8	1, 7	syl 18	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝜓)
9		optocl.1	. 2 ⊢ 𝐷 = (𝐵 × 𝐶)
10	8, 9	eleq2s 2880	1 ⊢ (𝐴 ∈ 𝐷 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569  ∃wex 1808   ∈ wcel 2142  ⟨cop 4594   × cxp 5658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-opab 5173  df-xp 5666

This theorem is used by:  2optocl  5756  3optocl  5757  ecoptocl  8803  ax1rid  11152  axcnre  11155