| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 4531 | . . . 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 4531 | . . 3 ⊢ (𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐷) | |
| 5 | 3, 4 | eqtr4d 2780 | . 2 ⊢ (𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸)) |
| 6 | iffalse 4534 | . . . 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 4534 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐸) | |
| 10 | 8, 9 | eqtr4d 2780 | . 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 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-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-if 4526 |
| This theorem is referenced by: fvif 6922 iffv 6923 ovif 7531 ovif2 7532 ifov 7534 xmulneg1 13311 efrlim 27012 efrlimOLD 27013 lgsneg 27365 lgsdilem 27368 rpvmasum2 27556 |
| Copyright terms: Public domain | W3C validator |