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Mirrors > Home > MPE Home > Th. List > eqif | Structured version Visualization version GIF version |
Description: Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.) |
Ref | Expression |
---|---|
eqif | ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2736 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ 𝐴 = 𝐵)) | |
2 | eqeq2 2736 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ 𝐴 = 𝐶)) | |
3 | 1, 2 | elimif 4558 | 1 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1533 ifcif 4521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-if 4522 |
This theorem is referenced by: ifval 4563 xpima 6172 fin23lem19 10328 fin23lem28 10332 fin23lem29 10333 fin23lem30 10334 aalioulem3 26190 ifnebib 32253 iocinif 32464 fsumcvg4 33422 ind1a 33509 esumsnf 33554 itg2addnclem2 37034 clsk1indlem4 43309 afvpcfv0 46364 |
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