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Mirrors > Home > MPE Home > Th. List > optocl | Structured version Visualization version GIF version |
Description: Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.) |
Ref | Expression |
---|---|
optocl.1 | ⊢ 𝐷 = (𝐵 × 𝐶) |
optocl.2 | ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓)) |
optocl.3 | ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑) |
Ref | Expression |
---|---|
optocl | ⊢ (𝐴 ∈ 𝐷 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp3 5703 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶))) | |
2 | opelxp 5674 | . . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) | |
3 | optocl.3 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑) | |
4 | 2, 3 | sylbi 216 | . . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) → 𝜑) |
5 | optocl.2 | . . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | 4, 5 | imbitrid 243 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) → 𝜓)) |
7 | 6 | imp 408 | . . . 4 ⊢ ((⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) → 𝜓) |
8 | 7 | exlimivv 1936 | . . 3 ⊢ (∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) → 𝜓) |
9 | 1, 8 | sylbi 216 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝜓) |
10 | optocl.1 | . 2 ⊢ 𝐷 = (𝐵 × 𝐶) | |
11 | 9, 10 | eleq2s 2856 | 1 ⊢ (𝐴 ∈ 𝐷 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ⟨cop 4597 × cxp 5636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-opab 5173 df-xp 5644 |
This theorem is referenced by: 2optocl 5732 3optocl 5733 ecoptocl 8753 ax1rid 11104 axcnre 11107 |
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