Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2633 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ 𝐴 = 𝐵)) | |
2 | eqeq2 2633 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ 𝐴 = 𝐶)) | |
3 | 1, 2 | elimif 4122 | 1 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ifcif 4086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-if 4087 |
This theorem is referenced by: ifval 4127 xpima 5576 fin23lem19 9158 fin23lem28 9162 fin23lem29 9163 fin23lem30 9164 aalioulem3 24089 iocinif 29543 fsumcvg4 29996 ind1a 30081 esumsnf 30126 itg2addnclem2 33462 clsk1indlem4 38342 afvpcfv0 41226 |
Copyright terms: Public domain | W3C validator |