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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 4481 | . . 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 4464 | . . 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 2766 | 1 ⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1539 ifcif 4459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3431 df-un 3891 df-if 4460 |
This theorem is referenced by: ifbieq12i 4486 gcdcom 16230 gcdass 16265 lcmcom 16308 lcmass 16329 bj-xpimasn 35153 cdleme31sdnN 38409 cdlemefr44 38447 cdleme48fv 38521 cdlemeg49lebilem 38561 cdleme50eq 38563 hoidmvlelem3 44116 hoidmvlelem4 44117 |
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