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Mirrors > Home > ILE Home > Th. List > mtbid | GIF version |
Description: A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.) |
Ref | Expression |
---|---|
mtbid.min | ⊢ (𝜑 → ¬ 𝜓) |
mtbid.maj | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
mtbid | ⊢ (𝜑 → ¬ 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mtbid.min | . 2 ⊢ (𝜑 → ¬ 𝜓) | |
2 | mtbid.maj | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 2 | biimprd 156 | . 2 ⊢ (𝜑 → (𝜒 → 𝜓)) |
4 | 1, 3 | mtod 624 | 1 ⊢ (𝜑 → ¬ 𝜒) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: sylnib 636 eqneltrrd 2184 neleqtrd 2185 eueq3dc 2789 efrirr 4180 fidcenumlemrks 6662 nqnq0pi 6997 zdclt 8824 frec2uzf1od 9813 expnegap0 9963 ibcval5 10171 zfz1isolemiso 10244 iseqcoll 10247 fisumss 10784 rpdvds 11359 oddpwdclemodd 11428 |
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