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Mirrors > Home > ILE Home > Th. List > syl6com | GIF version |
Description: Syllogism inference with commuted antecedents. (Contributed by NM, 25-May-2005.) |
Ref | Expression |
---|---|
syl6com.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
syl6com.2 | ⊢ (𝜒 → 𝜃) |
Ref | Expression |
---|---|
syl6com | ⊢ (𝜓 → (𝜑 → 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl6com.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | syl6com.2 | . . 3 ⊢ (𝜒 → 𝜃) | |
3 | 1, 2 | syl6 33 | . 2 ⊢ (𝜑 → (𝜓 → 𝜃)) |
4 | 3 | com12 30 | 1 ⊢ (𝜓 → (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: pclem6 1369 spimh 1730 ax16 1806 ax16i 1851 elres 4927 funcnvuni 5267 funrnex 6093 negf1o 8301 lidrididd 12636 dfgrp2 12732 basis2 12840 bj-inf2vnlem2 14006 |
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