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Mirrors > Home > ILE Home > Th. List > mtbi | GIF version |
Description: An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Oct-2012.) |
Ref | Expression |
---|---|
mtbi.1 | ⊢ ¬ 𝜑 |
mtbi.2 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
mtbi | ⊢ ¬ 𝜓 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mtbi.1 | . 2 ⊢ ¬ 𝜑 | |
2 | mtbi.2 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
3 | 2 | biimpri 132 | . 2 ⊢ (𝜓 → 𝜑) |
4 | 1, 3 | mto 651 | 1 ⊢ ¬ 𝜓 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: mtbir 660 vnex 4059 onsucelsucexmid 4445 dtruex 4474 dmsn0 5006 php5 6752 exmidonfinlem 7049 ndvdsi 11630 nprmi 11805 unennn 11910 bj-vprc 13094 bj-vnex 13096 |
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