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Mirrors > Home > MPE Home > Th. List > ifbieq12i | Structured version Visualization version GIF version |
Description: Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.) |
Ref | Expression |
---|---|
ifbieq12i.1 | ⊢ (𝜑 ↔ 𝜓) |
ifbieq12i.2 | ⊢ 𝐴 = 𝐶 |
ifbieq12i.3 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
ifbieq12i | ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifbieq12i.2 | . . 3 ⊢ 𝐴 = 𝐶 | |
2 | ifeq1 4040 | . . 3 ⊢ (𝐴 = 𝐶 → if(𝜑, 𝐴, 𝐵) = if(𝜑, 𝐶, 𝐵)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜑, 𝐶, 𝐵) |
4 | ifbieq12i.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
5 | ifbieq12i.3 | . . 3 ⊢ 𝐵 = 𝐷 | |
6 | 4, 5 | ifbieq2i 4060 | . 2 ⊢ if(𝜑, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷) |
7 | 3, 6 | eqtri 2632 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ifcif 4036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-un 3545 df-if 4037 |
This theorem is referenced by: cbvditg 23424 sgnneg 29929 binomcxplemdvsum 37576 |
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