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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 4496 | . . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
2 | 1 | con3i 154 | . . 3 ⊢ (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → ¬ 𝜑) |
3 | 2 | iffalsed 4501 | . 2 ⊢ (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → if(𝜑, 𝐴, 𝐵) = 𝐵) |
4 | 3 | orri 861 | 1 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 846 = wceq 1542 ifcif 4490 |
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 4491 |
This theorem is referenced by: ifpr 4656 rabrsn 4689 prmolefac 16926 muval2 26506 finxpreclem2 35911 relexpxpmin 42081 |
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