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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 5334 | . 2 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉 ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) | |
2 | eqcom 2805 | . 2 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ 〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉) | |
3 | eqcom 2805 | . . 3 ⊢ (𝐴 = 𝐶 ↔ 𝐶 = 𝐴) | |
4 | eqcom 2805 | . . 3 ⊢ (𝐵 = 𝐷 ↔ 𝐷 = 𝐵) | |
5 | 3, 4 | anbi12i 629 | . 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 1538 ∈ wcel 2111 〈cop 4531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 |
This theorem is referenced by: opth2 5337 fliftel 7041 symg2bas 18513 brsnop 30453 projf1o 41825 |
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