![]() |
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 4431 | . . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
2 | 1 | con3i 157 | . . 3 ⊢ (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → ¬ 𝜑) |
3 | 2 | iffalsed 4436 | . 2 ⊢ (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → if(𝜑, 𝐴, 𝐵) = 𝐵) |
4 | 3 | orri 859 | 1 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 844 = wceq 1538 ifcif 4425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-if 4426 |
This theorem is referenced by: ifpr 4589 rabrsn 4620 prmolefac 16372 muval2 25719 finxpreclem2 34807 relexpxpmin 40418 |
Copyright terms: Public domain | W3C validator |