![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
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 4639 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
2 | sneq 4639 | . . 3 ⊢ (𝐶 = 𝐷 → {𝐶} = {𝐷}) | |
3 | 1, 2 | anim12i 612 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ({𝐴} = {𝐵} ∧ {𝐶} = {𝐷})) |
4 | altopthsn 35234 | . 2 ⊢ (⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫ ↔ ({𝐴} = {𝐵} ∧ {𝐶} = {𝐷})) | |
5 | 3, 4 | sylibr 233 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 {csn 4629 ⟪caltop 35229 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-sn 4630 df-pr 4632 df-altop 35231 |
This theorem is referenced by: altopeq1 35236 altopeq2 35237 |
Copyright terms: Public domain | W3C validator |