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Mirrors > Home > MPE Home > Th. List > opthg2 | Structured version Visualization version GIF version |
Description: Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opthg2 | ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opthg 5376 | . 2 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉 ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) | |
2 | eqcom 2745 | . 2 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ 〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉) | |
3 | eqcom 2745 | . . 3 ⊢ (𝐴 = 𝐶 ↔ 𝐶 = 𝐴) | |
4 | eqcom 2745 | . . 3 ⊢ (𝐵 = 𝐷 ↔ 𝐷 = 𝐵) | |
5 | 3, 4 | anbi12i 630 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)) |
6 | 1, 2, 5 | 3bitr4g 317 | 1 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2111 〈cop 4562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pr 5337 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-rab 3071 df-v 3423 df-dif 3884 df-un 3886 df-nul 4253 df-if 4455 df-sn 4557 df-pr 4559 df-op 4563 |
This theorem is referenced by: opth2 5379 brsnop 5420 fliftel 7137 symg2bas 18810 projf1o 42438 |
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