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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 35970 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 2744 | . 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 35970 | . 2 ⊢ (⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫ ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)) |
| 5 | eqcom 2744 | . . 3 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶) | |
| 6 | eqcom 2744 | . . 3 ⊢ (𝐷 = 𝐵 ↔ 𝐵 = 𝐷) | |
| 7 | 5, 6 | anbi12i 628 | . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⟪caltop 35957 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 df-altop 35959 |
| This theorem is referenced by: (None) |
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