Theorem 2optocl 5119

Description: Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)

Hypotheses

Ref	Expression
2optocl.1	⊢ 𝑅 = (𝐶 × 𝐷)
2optocl.2	⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓))
2optocl.3	⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓 ↔ 𝜒))
2optocl.4	⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) → 𝜑)

Assertion

Ref	Expression
2optocl	⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜒)

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝐴   𝑧,𝐵,𝑤   𝑥,𝐶,𝑦,𝑧,𝑤   𝑥,𝐷,𝑦,𝑧,𝑤   𝜓,𝑥,𝑦   𝜒,𝑧,𝑤   𝑧,𝑅,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑧,𝑤)   𝜒(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem 2optocl

Step	Hyp	Ref	Expression
1		2optocl.1	. . 3 ⊢ 𝑅 = (𝐶 × 𝐷)
2		2optocl.3	. . . 4 ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓 ↔ 𝜒))
3	2	imbi2d 329	. . 3 ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → ((𝐴 ∈ 𝑅 → 𝜓) ↔ (𝐴 ∈ 𝑅 → 𝜒)))
4		2optocl.2	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓))
5	4	imbi2d 329	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → 𝜑) ↔ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → 𝜓)))
6		2optocl.4	. . . . . 6 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) → 𝜑)
7	6	ex 449	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → 𝜑))
8	1, 5, 7	optocl 5118	. . . 4 ⊢ (𝐴 ∈ 𝑅 → ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → 𝜓))
9	8	com12 32	. . 3 ⊢ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → (𝐴 ∈ 𝑅 → 𝜓))
10	1, 3, 9	optocl 5118	. 2 ⊢ (𝐵 ∈ 𝑅 → (𝐴 ∈ 𝑅 → 𝜒))
11	10	impcom 445	1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   × cxp 5036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044

This theorem is referenced by:  3optocl  5120  ecopovsym  7736  axaddf  9845  axmulf  9846