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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 35474 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 2733 | . 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 35475 | . 2 ⊢ (⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫ ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)) |
5 | eqcom 2733 | . . 3 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶) | |
6 | eqcom 2733 | . . 3 ⊢ (𝐷 = 𝐵 ↔ 𝐵 = 𝐷) | |
7 | 5, 6 | anbi12i 626 | . 2 ⊢ ((𝐶 = 𝐴 ∧ 𝐷 = 𝐵) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
8 | 1, 4, 7 | 3bitri 297 | 1 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⟪caltop 35461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-sn 4624 df-pr 4626 df-altop 35463 |
This theorem is referenced by: (None) |
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