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Mirrors > Home > ILE Home > Th. List > con2b | GIF version |
Description: Contraposition. Bidirectional version of con2 638. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
con2b | ⊢ ((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | con2 638 | . 2 ⊢ ((𝜑 → ¬ 𝜓) → (𝜓 → ¬ 𝜑)) | |
2 | con2 638 | . 2 ⊢ ((𝜓 → ¬ 𝜑) → (𝜑 → ¬ 𝜓)) | |
3 | 1, 2 | impbii 125 | 1 ⊢ ((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: mt2bi 679 pm4.15 689 dfdif3 3237 ssconb 3260 disjsn 3645 isprm3 12072 |
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