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Mirrors > Home > ILE Home > Th. List > sylan9 | GIF version |
Description: Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
Ref | Expression |
---|---|
sylan9.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
sylan9.2 | ⊢ (𝜃 → (𝜒 → 𝜏)) |
Ref | Expression |
---|---|
sylan9 | ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylan9.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | sylan9.2 | . . 3 ⊢ (𝜃 → (𝜒 → 𝜏)) | |
3 | 1, 2 | syl9 66 | . 2 ⊢ (𝜑 → (𝜃 → (𝜓 → 𝜏))) |
4 | 3 | imp 115 | 1 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜏)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 |
This theorem is referenced by: sbequi 1720 rspc2 2661 rspc3v 2665 trintssm 3870 copsexg 3981 chfnrn 5278 ffnfv 5323 f1elima 5412 smoel2 5918 th3q 6211 addnnnq0 6547 mulnnnq0 6548 addsrpr 6830 mulsrpr 6831 cau3lem 9710 |
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