![]() |
Mathbox for Wolf Lammen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-ifp-ncond2 | Structured version Visualization version GIF version |
Description: If one case of an if- condition is false, the other automatically follows. (Contributed by Wolf Lammen, 21-Jul-2024.) |
Ref | Expression |
---|---|
wl-ifp-ncond2 | ⊢ (¬ 𝜒 → (if-(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-ifp-ncond1 35146 | . 2 ⊢ (¬ 𝜒 → (if-(¬ 𝜑, 𝜒, 𝜓) ↔ (¬ ¬ 𝜑 ∧ 𝜓))) | |
2 | ifpn 1070 | . 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓)) | |
3 | notnotb 319 | . . 3 ⊢ (𝜑 ↔ ¬ ¬ 𝜑) | |
4 | 3 | anbi1i 627 | . 2 ⊢ ((𝜑 ∧ 𝜓) ↔ (¬ ¬ 𝜑 ∧ 𝜓)) |
5 | 1, 2, 4 | 3bitr4g 318 | 1 ⊢ (¬ 𝜒 → (if-(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 if-wif 1059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-ifp 1060 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |