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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 5500 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 2744 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ↔ 〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) | |
| 2 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 3 | vex 3484 | . . . . . 6 ⊢ 𝑦 ∈ V | |
| 4 | 2, 3 | opth 5481 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) |
| 5 | 1, 4 | bitri 275 | . . . 4 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) |
| 6 | 5 | anbi1i 624 | . . 3 ⊢ ((〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑)) |
| 7 | 6 | 2exbii 1849 | . 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 3527 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ 𝜓)) |
| 11 | 7, 10 | bitrid 283 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 〈cop 4632 |
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 |
| This theorem is referenced by: opelopabga 5538 ov6g 7597 ltresr 11180 |
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