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Mirrors > Home > MPE Home > Th. List > Mathboxes > ifpnot23c | Structured version Visualization version GIF version |
Description: Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.) |
Ref | Expression |
---|---|
ifpnot23c | ⊢ (¬ if-(𝜑, 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ 𝜓, 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifpnot23 40587 | . 2 ⊢ (¬ if-(𝜑, 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ ¬ 𝜒)) | |
2 | notnotb 318 | . . 3 ⊢ (𝜒 ↔ ¬ ¬ 𝜒) | |
3 | ifpbi3 40577 | . . 3 ⊢ ((𝜒 ↔ ¬ ¬ 𝜒) → (if-(𝜑, ¬ 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ ¬ 𝜒))) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (if-(𝜑, ¬ 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ ¬ 𝜒)) |
5 | 1, 4 | bitr4i 281 | 1 ⊢ (¬ if-(𝜑, 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ 𝜓, 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 if-wif 1058 |
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 400 df-or 845 df-ifp 1059 |
This theorem is referenced by: ifpnim1 40606 ifpnim2 40608 |
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