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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 4551 | . . 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 4534 | . . 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 2756 | 1 ⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ifcif 4529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-un 3952 df-if 4530 |
This theorem is referenced by: ifbieq12i 4556 gcdcom 16487 gcdass 16522 lcmcom 16563 lcmass 16584 bj-xpimasn 36434 cdleme31sdnN 39860 cdlemefr44 39898 cdleme48fv 39972 cdlemeg49lebilem 40012 cdleme50eq 40014 hoidmvlelem3 45985 hoidmvlelem4 45986 |
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