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Mirrors > Home > MPE Home > Th. List > syl3an1b | Structured version Visualization version GIF version |
Description: A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
Ref | Expression |
---|---|
syl3an1b.1 | ⊢ (𝜑 ↔ 𝜓) |
syl3an1b.2 | ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
syl3an1b | ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3an1b.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | biimpi 218 | . 2 ⊢ (𝜑 → 𝜓) |
3 | syl3an1b.2 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) | |
4 | 2, 3 | syl3an1 1159 | 1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 |
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 209 df-an 399 df-3an 1085 |
This theorem is referenced by: ovmpoelrn 7764 irrmul 12367 xrlttr 12527 flfneii 22594 padct 30449 crefdf 31107 divrngcl 35229 |
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