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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 4140 | . . 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 4124 | . . 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 2673 | 1 ⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1523 ifcif 4119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-rab 2950 df-v 3233 df-un 3612 df-if 4120 |
This theorem is referenced by: ifbieq12i 4145 gcdcom 15282 gcdass 15311 lcmcom 15353 lcmass 15374 bj-xpimasn 33067 cdleme31sdnN 35992 cdlemefr44 36030 cdleme48fv 36104 cdlemeg49lebilem 36144 cdleme50eq 36146 hoidmvlelem3 41132 hoidmvlelem4 41133 |
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