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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 4640 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
2 | sneq 4640 | . . 3 ⊢ (𝐶 = 𝐷 → {𝐶} = {𝐷}) | |
3 | 1, 2 | anim12i 613 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ({𝐴} = {𝐵} ∧ {𝐶} = {𝐷})) |
4 | altopthsn 35942 | . 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 1536 {csn 4630 ⟪caltop 35937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-sn 4631 df-pr 4633 df-altop 35939 |
This theorem is referenced by: altopeq1 35944 altopeq2 35945 |
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