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Mirrors > Home > ILE Home > Th. List > sylan9bbr | GIF version |
Description: Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.) |
Ref | Expression |
---|---|
sylan9bbr.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
sylan9bbr.2 | ⊢ (𝜃 → (𝜒 ↔ 𝜏)) |
Ref | Expression |
---|---|
sylan9bbr | ⊢ ((𝜃 ∧ 𝜑) → (𝜓 ↔ 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylan9bbr.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | sylan9bbr.2 | . . 3 ⊢ (𝜃 → (𝜒 ↔ 𝜏)) | |
3 | 1, 2 | sylan9bb 449 | . 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜏)) |
4 | 3 | ancoms 264 | 1 ⊢ ((𝜃 ∧ 𝜑) → (𝜓 ↔ 𝜏)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: pm5.75 903 mpteq12f 3858 opelopabsb 4015 elreimasng 4711 fvelrnb 5242 fmptco 5351 fconstfvm 5400 f1oiso 5485 mpt2eq123 5584 dfoprab4f 5839 fmpt2x 5846 nnmword 6114 ltmpig 6529 qreccl 8727 0fz1 9064 zmodid2 9354 divgcdcoprm0 10483 cbvrald 10598 |
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