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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 31246 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 2617 | . 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 31246 | . 2 ⊢ (⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫ ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)) |
5 | eqcom 2617 | . . 3 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶) | |
6 | eqcom 2617 | . . 3 ⊢ (𝐷 = 𝐵 ↔ 𝐵 = 𝐷) | |
7 | 5, 6 | anbi12i 729 | . 2 ⊢ ((𝐶 = 𝐴 ∧ 𝐷 = 𝐵) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
8 | 1, 4, 7 | 3bitri 285 | 1 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⟪caltop 31233 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 df-pr 4128 df-altop 31235 |
This theorem is referenced by: (None) |
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