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Mirrors > Home > MPE Home > Th. List > sylanr2 | Structured version Visualization version GIF version |
Description: A syllogism inference. (Contributed by NM, 9-Apr-2005.) |
Ref | Expression |
---|---|
sylanr2.1 | ⊢ (𝜑 → 𝜃) |
sylanr2.2 | ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏) |
Ref | Expression |
---|---|
sylanr2 | ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜑)) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylanr2.1 | . . 3 ⊢ (𝜑 → 𝜃) | |
2 | 1 | anim2i 616 | . 2 ⊢ ((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜃)) |
3 | sylanr2.2 | . 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏) | |
4 | 2, 3 | sylan2 592 | 1 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜑)) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 208 df-an 397 |
This theorem is referenced by: adantrrl 720 adantrrr 721 isfin7-2 9664 mulsub 10931 fzsubel 12793 expsub 13327 ramlb 16184 0ram 16185 ressmplvsca 19927 tgcl 21261 fgss2 22166 nmoid 23034 numclwwlkqhash 27846 chirredlem4 29861 pibt2 34229 lindsadd 34416 poimirlem28 34451 pridlc3 34883 stoweidlem34 41861 |
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