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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 4244 | . . 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 4228 | . . 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 2792 | 1 ⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1630 ifcif 4223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-rab 3069 df-v 3351 df-un 3726 df-if 4224 |
This theorem is referenced by: ifbieq12i 4249 gcdcom 15442 gcdass 15471 lcmcom 15513 lcmass 15534 bj-xpimasn 33267 cdleme31sdnN 36189 cdlemefr44 36227 cdleme48fv 36301 cdlemeg49lebilem 36341 cdleme50eq 36343 hoidmvlelem3 41325 hoidmvlelem4 41326 |
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