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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 2622 | . 2 ⊢ 𝐶 = 𝐶 | |
2 | altopeq12 32069 | . 2 ⊢ ((𝐶 = 𝐶 ∧ 𝐴 = 𝐵) → ⟪𝐶, 𝐴⟫ = ⟪𝐶, 𝐵⟫) | |
3 | 1, 2 | mpan 706 | 1 ⊢ (𝐴 = 𝐵 → ⟪𝐶, 𝐴⟫ = ⟪𝐶, 𝐵⟫) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ⟪caltop 32063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-sn 4178 df-pr 4180 df-altop 32065 |
This theorem is referenced by: sbcaltop 32088 |
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