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Mirrors > Home > ILE Home > Th. List > syl2im | GIF version |
Description: Replace two antecedents. Implication-only version of syl2an 287. (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 32 | . 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-mp 5 ax-1 6 ax-2 7 |
This theorem is referenced by: syl2imc 39 sylc 62 bi3ant 223 pm3.12dc 943 pm3.13dc 944 nfrimi 1506 abnex 4376 vtoclr 4595 funopg 5165 xpider 6508 ixxssixx 9715 difelfzle 9942 txcnp 12479 bj-inf2vnlem1 13339 |
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