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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 5169 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(〈𝑥, 𝑦〉 = 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶))) | |
2 | opelxp 5146 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) | |
3 | optocl.3 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑) | |
4 | 2, 3 | sylbi 207 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶) → 𝜑) |
5 | optocl.2 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | 4, 5 | syl5ib 234 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶) → 𝜓)) |
7 | 6 | imp 445 | . . . 4 ⊢ ((〈𝑥, 𝑦〉 = 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶)) → 𝜓) |
8 | 7 | exlimivv 1860 | . . 3 ⊢ (∃𝑥∃𝑦(〈𝑥, 𝑦〉 = 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶)) → 𝜓) |
9 | 1, 8 | sylbi 207 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝜓) |
10 | optocl.1 | . 2 ⊢ 𝐷 = (𝐵 × 𝐶) | |
11 | 9, 10 | eleq2s 2719 | 1 ⊢ (𝐴 ∈ 𝐷 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 〈cop 4183 × cxp 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 df-xp 5120 |
This theorem is referenced by: 2optocl 5196 3optocl 5197 ecoptocl 7837 ax1rid 9982 axcnre 9985 |
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