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Mirrors > Home > MPE Home > Th. List > copsex2g | Structured version Visualization version GIF version |
Description: Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.) Use a similar proof to copsex4g 5497 to reduce axiom usage. (Revised by SN, 1-Sep-2024.) |
Ref | Expression |
---|---|
copsex2g.1 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
copsex2g | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2732 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ↔ 〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) | |
2 | vex 3465 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3465 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opth 5478 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) |
5 | 1, 4 | bitri 274 | . . . 4 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) |
6 | 5 | anbi1i 622 | . . 3 ⊢ ((〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑)) |
7 | 6 | 2exbii 1843 | . 2 ⊢ (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑)) |
8 | id 22 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) | |
9 | copsex2g.1 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
10 | 8, 9 | cgsex2g 3508 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ 𝜓)) |
11 | 7, 10 | bitrid 282 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 〈cop 4636 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
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 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 |
This theorem is referenced by: opelopabga 5535 ov6g 7585 ltresr 11165 |
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