Theorem 3optocl 5756

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

Hypotheses

Ref	Expression
3optocl.1	⊢ 𝑅 = (𝐷 × 𝐹)
3optocl.2	⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓))
3optocl.3	⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓 ↔ 𝜒))
3optocl.4	⊢ (⟨𝑣, 𝑢⟩ = 𝐶 → (𝜒 ↔ 𝜃))
3optocl.5	⊢ (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐹) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐹) ∧ (𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹)) → 𝜑)

Assertion

Ref	Expression
3optocl	⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑅) → 𝜃)

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

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

Proof of Theorem 3optocl

Step	Hyp	Ref	Expression
1		3optocl.1	. . . 4 ⊢ 𝑅 = (𝐷 × 𝐹)
2		3optocl.4	. . . . 5 ⊢ (⟨𝑣, 𝑢⟩ = 𝐶 → (𝜒 ↔ 𝜃))
3	2	imbi2d 340	. . . 4 ⊢ (⟨𝑣, 𝑢⟩ = 𝐶 → (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜒) ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜃)))
4		3optocl.2	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓))
5	4	imbi2d 340	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (((𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹) → 𝜑) ↔ ((𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹) → 𝜓)))
6		3optocl.3	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓 ↔ 𝜒))
7	6	imbi2d 340	. . . . . 6 ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → (((𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹) → 𝜓) ↔ ((𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹) → 𝜒)))
8		3optocl.5	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐹) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐹) ∧ (𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹)) → 𝜑)
9	8	3expia 1121	. . . . . 6 ⊢ (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐹) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐹)) → ((𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹) → 𝜑))
10	1, 5, 7, 9	2optocl 5755	. . . . 5 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → ((𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹) → 𝜒))
11	10	com12 32	. . . 4 ⊢ ((𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹) → ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜒))
12	1, 3, 11	optocl 5754	. . 3 ⊢ (𝐶 ∈ 𝑅 → ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜃))
13	12	impcom 407	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) ∧ 𝐶 ∈ 𝑅) → 𝜃)
14	13	3impa 1109	1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑅) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ⟨cop 4612   × cxp 5657

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 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-opab 5187  df-xp 5665

This theorem is referenced by:  ecopovtrn  8839