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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 4577 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
2 | sneq 4577 | . . 3 ⊢ (𝐶 = 𝐷 → {𝐶} = {𝐷}) | |
3 | 1, 2 | anim12i 614 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ({𝐴} = {𝐵} ∧ {𝐶} = {𝐷})) |
4 | altopthsn 33422 | . 2 ⊢ (⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫ ↔ ({𝐴} = {𝐵} ∧ {𝐶} = {𝐷})) | |
5 | 3, 4 | sylibr 236 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 {csn 4567 ⟪caltop 33417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-sn 4568 df-pr 4570 df-altop 33419 |
This theorem is referenced by: altopeq1 33424 altopeq2 33425 |
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