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Mirrors > Home > MPE Home > Th. List > ifpn | Structured version Visualization version GIF version |
Description: Conditional operator for the negation of a proposition. (Contributed by BJ, 30-Sep-2019.) |
Ref | Expression |
---|---|
ifpn | ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnotb 304 | . . . 4 ⊢ (𝜑 ↔ ¬ ¬ 𝜑) | |
2 | 1 | imbi1i 339 | . . 3 ⊢ ((𝜑 → 𝜓) ↔ (¬ ¬ 𝜑 → 𝜓)) |
3 | 2 | anbi2ci 732 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)) ↔ ((¬ 𝜑 → 𝜒) ∧ (¬ ¬ 𝜑 → 𝜓))) |
4 | dfifp2 1014 | . 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) | |
5 | dfifp2 1014 | . 2 ⊢ (if-(¬ 𝜑, 𝜒, 𝜓) ↔ ((¬ 𝜑 → 𝜒) ∧ (¬ ¬ 𝜑 → 𝜓))) | |
6 | 3, 4, 5 | 3bitr4i 292 | 1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 if-wif 1012 |
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 197 df-or 385 df-an 386 df-ifp 1013 |
This theorem is referenced by: ifpfal 1024 ifpdfbi 37818 ifpxorcor 37821 |
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