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Mirrors > Home > ILE Home > Th. List > syl6an | GIF version |
Description: A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.) |
Ref | Expression |
---|---|
syl6an.1 | ⊢ (𝜑 → 𝜓) |
syl6an.2 | ⊢ (𝜑 → (𝜒 → 𝜃)) |
syl6an.3 | ⊢ ((𝜓 ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
syl6an | ⊢ (𝜑 → (𝜒 → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl6an.2 | . . 3 ⊢ (𝜑 → (𝜒 → 𝜃)) | |
2 | syl6an.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
3 | 1, 2 | jctild 299 | . 2 ⊢ (𝜑 → (𝜒 → (𝜓 ∧ 𝜃))) |
4 | syl6an.3 | . 2 ⊢ ((𝜓 ∧ 𝜃) → 𝜏) | |
5 | 3, 4 | syl6 29 | 1 ⊢ (𝜑 → (𝜒 → 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-ia3 101 |
This theorem is referenced by: rdgon 5973 prarloclem5 6598 ltsopr 6694 nominpos 8162 ublbneg 8548 absle 9685 climshftlemg 9823 serif0 9871 bj-indind 10056 |
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