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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 2746 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ 𝐴 = 𝐵)) | |
2 | eqeq2 2746 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ 𝐴 = 𝐶)) | |
3 | 1, 2 | elimif 4567 | 1 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1536 ifcif 4530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-if 4531 |
This theorem is referenced by: ifval 4572 xpima 6203 fin23lem19 10373 fin23lem28 10377 fin23lem29 10378 fin23lem30 10379 aalioulem3 26390 ifnebib 32569 iocinif 32789 fsumcvg4 33910 ind1a 33999 esumsnf 34044 itg2addnclem2 37658 clsk1indlem4 44033 afvpcfv0 47095 |
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