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Mirrors > Home > ILE Home > Th. List > syl2im | GIF version |
Description: Replace two antecedents. Implication-only version of syl2an 273. (Contributed by Wolf Lammen, 14-May-2013.) |
Ref | Expression |
---|---|
syl2im.1 | ⊢ (𝜑 → 𝜓) |
syl2im.2 | ⊢ (𝜒 → 𝜃) |
syl2im.3 | ⊢ (𝜓 → (𝜃 → 𝜏)) |
Ref | Expression |
---|---|
syl2im | ⊢ (𝜑 → (𝜒 → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl2im.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | syl2im.2 | . . 3 ⊢ (𝜒 → 𝜃) | |
3 | syl2im.3 | . . 3 ⊢ (𝜓 → (𝜃 → 𝜏)) | |
4 | 2, 3 | syl5 28 | . 2 ⊢ (𝜓 → (𝜒 → 𝜏)) |
5 | 1, 4 | syl 14 | 1 ⊢ (𝜑 → (𝜒 → 𝜏)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 |
This theorem is referenced by: sylc 56 bi3ant 213 pm3.12dc 865 pm3.13dc 866 nfrimi 1418 vtoclr 4388 funopg 4934 xpiderm 6177 ixxssixx 8771 difelfzle 8992 bj-inf2vnlem1 10095 |
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