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Mirrors > Home > ILE Home > Th. List > nbn2 | GIF version |
Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Ref | Expression |
---|---|
nbn2 | ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜑 ↔ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.21im 612 | . 2 ⊢ (¬ 𝜑 → (¬ 𝜓 → (𝜑 ↔ 𝜓))) | |
2 | bi2 121 | . . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) | |
3 | mtt 610 | . . 3 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜓 → 𝜑))) | |
4 | 2, 3 | syl5ibr 145 | . 2 ⊢ (¬ 𝜑 → ((𝜑 ↔ 𝜓) → ¬ 𝜓)) |
5 | 1, 4 | impbid 120 | 1 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜑 ↔ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: bibif 614 pm5.18dc 777 biassdc 1286 |
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