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Mirrors > Home > MPE Home > Th. List > ifsb | Structured version Visualization version GIF version |
Description: Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.) |
Ref | Expression |
---|---|
ifsb.1 | ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝐶 = 𝐷) |
ifsb.2 | ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → 𝐶 = 𝐸) |
Ref | Expression |
---|---|
ifsb | ⊢ 𝐶 = if(𝜑, 𝐷, 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 4535 | . . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
2 | ifsb.1 | . . . 4 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝐶 = 𝐷) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) |
4 | iftrue 4535 | . . 3 ⊢ (𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐷) | |
5 | 3, 4 | eqtr4d 2776 | . 2 ⊢ (𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸)) |
6 | iffalse 4538 | . . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
7 | ifsb.2 | . . . 4 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → 𝐶 = 𝐸) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (¬ 𝜑 → 𝐶 = 𝐸) |
9 | iffalse 4538 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐸) | |
10 | 8, 9 | eqtr4d 2776 | . 2 ⊢ (¬ 𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸)) |
11 | 5, 10 | pm2.61i 182 | 1 ⊢ 𝐶 = if(𝜑, 𝐷, 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ifcif 4529 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-if 4530 |
This theorem is referenced by: fvif 6908 iffv 6909 ovif 7506 ovif2 7507 ifov 7509 xmulneg1 13248 efrlim 26474 lgsneg 26824 lgsdilem 26827 rpvmasum2 27015 |
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