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Mirrors > Home > MPE Home > Th. List > Mathboxes > ifpnot23d | Structured version Visualization version GIF version |
Description: Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.) |
Ref | Expression |
---|---|
ifpnot23d | ⊢ (¬ if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(𝜑, 𝜓, 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifpnot23 41062 | . 2 ⊢ (¬ if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ ¬ 𝜒)) | |
2 | notnotb 315 | . . 3 ⊢ (𝜓 ↔ ¬ ¬ 𝜓) | |
3 | notnotb 315 | . . 3 ⊢ (𝜒 ↔ ¬ ¬ 𝜒) | |
4 | ifpbi23 41057 | . . 3 ⊢ (((𝜓 ↔ ¬ ¬ 𝜓) ∧ (𝜒 ↔ ¬ ¬ 𝜒)) → (if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ ¬ 𝜒))) | |
5 | 2, 3, 4 | mp2an 689 | . 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ ¬ 𝜒)) |
6 | 1, 5 | bitr4i 277 | 1 ⊢ (¬ if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(𝜑, 𝜓, 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 if-wif 1060 |
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 206 df-an 397 df-or 845 df-ifp 1061 |
This theorem is referenced by: ifpororb 41089 |
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