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Mathbox for Jarvin Udandy |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > abciffcbatnabciffncbai | Structured version Visualization version GIF version |
Description: Operands in a biconditional expression converted negated. Additionally biconditional converted to show antecedent implies sequent. (Contributed by Jarvin Udandy, 7-Sep-2020.) |
Ref | Expression |
---|---|
abciffcbatnabciffncbai.1 | ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜒 ∧ 𝜓) ∧ 𝜑)) |
Ref | Expression |
---|---|
abciffcbatnabciffncbai | ⊢ (¬ ((𝜑 ∧ 𝜓) ∧ 𝜒) → ¬ ((𝜒 ∧ 𝜓) ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abciffcbatnabciffncbai.1 | . . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜒 ∧ 𝜓) ∧ 𝜑)) | |
2 | notbi 308 | . . . 4 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜒 ∧ 𝜓) ∧ 𝜑)) ↔ (¬ ((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ¬ ((𝜒 ∧ 𝜓) ∧ 𝜑))) | |
3 | 2 | biimpi 205 | . . 3 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜒 ∧ 𝜓) ∧ 𝜑)) → (¬ ((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ¬ ((𝜒 ∧ 𝜓) ∧ 𝜑))) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ (¬ ((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ¬ ((𝜒 ∧ 𝜓) ∧ 𝜑)) |
5 | 4 | biimpi 205 | 1 ⊢ (¬ ((𝜑 ∧ 𝜓) ∧ 𝜒) → ¬ ((𝜒 ∧ 𝜓) ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 |
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 196 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |