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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 2810 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ 𝐴 = 𝐵)) | |
2 | eqeq2 2810 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ 𝐴 = 𝐶)) | |
3 | 1, 2 | elimif 4461 | 1 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ∨ 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: ifval 4466 xpima 6006 fin23lem19 9747 fin23lem28 9751 fin23lem29 9752 fin23lem30 9753 aalioulem3 24930 iocinif 30530 fsumcvg4 31303 ind1a 31388 esumsnf 31433 itg2addnclem2 35109 clsk1indlem4 40747 afvpcfv0 43702 |
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