| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > ifeqor | Structured version Visualization version GIF version | ||
| Description: The possible values of a conditional operator. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
| Ref | Expression |
|---|---|
| ifeqor | ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iftrue 4531 | . . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
| 2 | 1 | con3i 154 | . . 3 ⊢ (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → ¬ 𝜑) |
| 3 | 2 | iffalsed 4536 | . 2 ⊢ (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → if(𝜑, 𝐴, 𝐵) = 𝐵) |
| 4 | 3 | orri 863 | 1 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∨ wo 848 = 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: ifpr 4693 rabrsn 4724 prmolefac 17084 muval2 27177 abssor 28270 finxpreclem2 37391 relexpxpmin 43730 |
| Copyright terms: Public domain | W3C validator |