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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 4429 | . . 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 4450 | . 2 ⊢ if(𝜑, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷) |
7 | 3, 6 | eqtri 2759 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ifcif 4425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-un 3858 df-if 4426 |
This theorem is referenced by: cbvditg 24705 sgnneg 32173 nosupcbv 33591 noinfcbv 33606 binomcxplemdvsum 41587 |
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