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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 4548 | . . 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 4530 | . . 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 2765 | 1 ⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ifcif 4525 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-un 3956 df-if 4526 |
| This theorem is referenced by: ifbieq12i 4553 gcdcom 16550 gcdass 16584 lcmcom 16630 lcmass 16651 bj-xpimasn 36956 cdleme31sdnN 40389 cdlemefr44 40427 cdleme48fv 40501 cdlemeg49lebilem 40541 cdleme50eq 40543 redvmptabs 42390 hoidmvlelem3 46612 hoidmvlelem4 46613 |
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