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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 4470 | . . 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 4490 | . 2 ⊢ if(𝜑, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷) |
7 | 3, 6 | eqtri 2844 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ifcif 4466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-un 3940 df-if 4467 |
This theorem is referenced by: cbvditg 24451 sgnneg 31798 binomcxplemdvsum 40685 |
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