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Mirrors > Home > MPE Home > Th. List > Mathboxes > altopthc | Structured version Visualization version GIF version |
Description: Alternate ordered pair theorem with different sethood requirements. See altopth 34936 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.) |
Ref | Expression |
---|---|
altopthc.1 | ⊢ 𝐵 ∈ V |
altopthc.2 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
altopthc | ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2739 | . 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫) | |
2 | altopthc.2 | . . 3 ⊢ 𝐶 ∈ V | |
3 | altopthc.1 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | altopthb 34937 | . 2 ⊢ (⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫ ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)) |
5 | eqcom 2739 | . . 3 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶) | |
6 | eqcom 2739 | . . 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 1541 ∈ wcel 2106 Vcvv 3474 ⟪caltop 34923 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-sn 4629 df-pr 4631 df-altop 34925 |
This theorem is referenced by: (None) |
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