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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 5407 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 2745 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ↔ 〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) | |
2 | vex 3433 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3433 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opth 5389 | . . . . 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 624 | . . 3 ⊢ ((〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑)) |
7 | 6 | 2exbii 1851 | . 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 3472 | . 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 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 〈cop 4567 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pr 5350 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3431 df-dif 3889 df-un 3891 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 |
This theorem is referenced by: opelopabga 5443 ov6g 7426 ltresr 10906 |
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