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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 147 | . 2 ⊢ (𝜑 → (𝜒 → 𝜓)) |
4 | 1, 3 | mtod 589 | 1 ⊢ (𝜑 → ¬ 𝜒) |
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: sylnib 601 eqneltrrd 2134 neleqtrd 2135 eueq3dc 2715 efrirr 4090 nqnq0pi 6536 zdclt 8318 frec2uzf1od 9192 expnegap0 9263 |
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