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Mirrors > Home > ILE Home > Th. List > mpisyl | GIF version |
Description: A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.) |
Ref | Expression |
---|---|
mpisyl.1 | ⊢ (𝜑 → 𝜓) |
mpisyl.2 | ⊢ 𝜒 |
mpisyl.3 | ⊢ (𝜓 → (𝜒 → 𝜃)) |
Ref | Expression |
---|---|
mpisyl | ⊢ (𝜑 → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpisyl.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | mpisyl.2 | . . 3 ⊢ 𝜒 | |
3 | mpisyl.3 | . . 3 ⊢ (𝜓 → (𝜒 → 𝜃)) | |
4 | 2, 3 | mpi 15 | . 2 ⊢ (𝜓 → 𝜃) |
5 | 1, 4 | syl 14 | 1 ⊢ (𝜑 → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 |
This theorem is referenced by: ceqsex 2775 reusv1 4458 fliftcnv 5795 fliftfun 5796 tfrlemibfn 6328 tfr1onlembfn 6344 tfrcllembfn 6357 cnvct 6808 ordiso 7034 exmidomni 7139 djudoml 7217 djudomr 7218 uzsinds 10441 fimaxq 10806 ltoddhalfle 11897 phicl2 12213 strsetsid 12494 txdis1cn 13748 xmeter 13906 subctctexmid 14720 |
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