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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 2749 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ 𝐴 = 𝐵)) | |
| 2 | eqeq2 2749 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ 𝐴 = 𝐶)) | |
| 3 | 1, 2 | elimif 4563 | 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 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: ifval 4568 xpima 6202 fin23lem19 10376 fin23lem28 10380 fin23lem29 10381 fin23lem30 10382 aalioulem3 26376 ifnebib 32562 iocinif 32783 ind1a 32844 fsumcvg4 33949 esumsnf 34065 itg2addnclem2 37679 clsk1indlem4 44057 afvpcfv0 47158 |
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