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Mirrors > Home > ILE Home > Th. List > syl3an3b | GIF version |
Description: A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
Ref | Expression |
---|---|
syl3an3b.1 | ⊢ (𝜑 ↔ 𝜃) |
syl3an3b.2 | ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
syl3an3b | ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3an3b.1 | . . 3 ⊢ (𝜑 ↔ 𝜃) | |
2 | 1 | biimpi 119 | . 2 ⊢ (𝜑 → 𝜃) |
3 | syl3an3b.2 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) | |
4 | 2, 3 | syl3an3 1268 | 1 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-3an 975 |
This theorem is referenced by: fvun2 5563 nnmsucr 6467 apreim 8522 xrlttr 9752 xrltso 9753 iccdil 9955 icccntr 9957 absexpzap 11044 unicld 12910 |
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