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Mirrors > Home > ILE Home > Th. List > mt2bi | GIF version |
Description: A false consequent falsifies an antecedent. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.) |
Ref | Expression |
---|---|
mt2bi.1 | ⊢ 𝜑 |
Ref | Expression |
---|---|
mt2bi | ⊢ (¬ 𝜓 ↔ (𝜓 → ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mt2bi.1 | . . 3 ⊢ 𝜑 | |
2 | 1 | a1bi 232 | . 2 ⊢ (¬ 𝜓 ↔ (𝜑 → ¬ 𝜓)) |
3 | con2b 593 | . 2 ⊢ ((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑)) | |
4 | 2, 3 | bitri 173 | 1 ⊢ (¬ 𝜓 ↔ (𝜓 → ¬ 𝜑)) |
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: (None) |
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