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| Mirrors > Home > MPE Home > Th. List > Mathboxes > altopeq12 | Structured version Visualization version GIF version | ||
| Description: Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.) |
| Ref | Expression |
|---|---|
| altopeq12 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sneq 4636 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
| 2 | sneq 4636 | . . 3 ⊢ (𝐶 = 𝐷 → {𝐶} = {𝐷}) | |
| 3 | 1, 2 | anim12i 613 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ({𝐴} = {𝐵} ∧ {𝐶} = {𝐷})) |
| 4 | altopthsn 35962 | . 2 ⊢ (⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫ ↔ ({𝐴} = {𝐵} ∧ {𝐶} = {𝐷})) | |
| 5 | 3, 4 | sylibr 234 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 {csn 4626 ⟪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: altopeq1 35964 altopeq2 35965 |
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