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Mirrors > Home > MPE Home > Th. List > Mathboxes > altopthd | Structured version Visualization version GIF version |
Description: Alternate ordered pair theorem with different sethood requirements. See altopth 34367 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.) |
Ref | Expression |
---|---|
altopthd.1 | ⊢ 𝐶 ∈ V |
altopthd.2 | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
altopthd | ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2743 | . 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫) | |
2 | altopthd.1 | . . 3 ⊢ 𝐶 ∈ V | |
3 | altopthd.2 | . . 3 ⊢ 𝐷 ∈ V | |
4 | 2, 3 | altopth 34367 | . 2 ⊢ (⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫ ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)) |
5 | eqcom 2743 | . . 3 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶) | |
6 | eqcom 2743 | . . 3 ⊢ (𝐷 = 𝐵 ↔ 𝐵 = 𝐷) | |
7 | 5, 6 | anbi12i 627 | . 2 ⊢ ((𝐶 = 𝐴 ∧ 𝐷 = 𝐵) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
8 | 1, 4, 7 | 3bitri 296 | 1 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 Vcvv 3441 ⟪caltop 34354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-sn 4574 df-pr 4576 df-altop 34356 |
This theorem is referenced by: (None) |
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