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Mirrors > Home > MPE Home > Th. List > ifbieq2i | Structured version Visualization version GIF version |
Description: Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
ifbieq2i.1 | ⊢ (𝜑 ↔ 𝜓) |
ifbieq2i.2 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
ifbieq2i | ⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifbieq2i.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | ifbi 4446 | . . 3 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐴) |
4 | ifbieq2i.2 | . . 3 ⊢ 𝐴 = 𝐵 | |
5 | ifeq2 4430 | . . 3 ⊢ (𝐴 = 𝐵 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵) |
7 | 3, 6 | eqtri 2821 | 1 ⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ifcif 4425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-un 3886 df-if 4426 |
This theorem is referenced by: ifbieq12i 4451 gcdcom 15852 gcdass 15885 lcmcom 15927 lcmass 15948 bj-xpimasn 34391 cdleme31sdnN 37683 cdlemefr44 37721 cdleme48fv 37795 cdlemeg49lebilem 37835 cdleme50eq 37837 hoidmvlelem3 43236 hoidmvlelem4 43237 |
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