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Mirrors > Home > MPE Home > Th. List > Mathboxes > altopeq2 | Structured version Visualization version GIF version |
Description: Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.) |
Ref | Expression |
---|---|
altopeq2 | ⊢ (𝐴 = 𝐵 → ⟪𝐶, 𝐴⟫ = ⟪𝐶, 𝐵⟫) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . 2 ⊢ 𝐶 = 𝐶 | |
2 | altopeq12 33425 | . 2 ⊢ ((𝐶 = 𝐶 ∧ 𝐴 = 𝐵) → ⟪𝐶, 𝐴⟫ = ⟪𝐶, 𝐵⟫) | |
3 | 1, 2 | mpan 688 | 1 ⊢ (𝐴 = 𝐵 → ⟪𝐶, 𝐴⟫ = ⟪𝐶, 𝐵⟫) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ⟪caltop 33419 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-sn 4570 df-pr 4572 df-altop 33421 |
This theorem is referenced by: sbcaltop 33444 |
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