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Mirrors > Home > MPE Home > Th. List > 2optocl | Structured version Visualization version GIF version |
Description: Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.) |
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) |
Ref | Expression |
---|---|
2optocl | ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2optocl.1 | . . 3 ⊢ 𝑅 = (𝐶 × 𝐷) | |
2 | 2optocl.3 | . . . 4 ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓 ↔ 𝜒)) | |
3 | 2 | imbi2d 339 | . . 3 ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → ((𝐴 ∈ 𝑅 → 𝜓) ↔ (𝐴 ∈ 𝑅 → 𝜒))) |
4 | 2optocl.2 | . . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | imbi2d 339 | . . . . 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 411 | . . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → 𝜑)) |
8 | 1, 5, 7 | optocl 5774 | . . . 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 5774 | . 2 ⊢ (𝐵 ∈ 𝑅 → (𝐴 ∈ 𝑅 → 𝜒)) |
11 | 10 | impcom 406 | 1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⟨cop 4636 × cxp 5678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-opab 5213 df-xp 5686 |
This theorem is referenced by: 3optocl 5776 ecopovsym 8842 axaddf 11174 axmulf 11175 |
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