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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 455 | . 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜏)) |
4 | 3 | ancoms 266 | 1 ⊢ ((𝜃 ∧ 𝜑) → (𝜓 ↔ 𝜏)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: pm5.75 927 mpteq12f 3966 opelopabsb 4140 elreimasng 4861 fvelrnb 5421 fmptco 5538 fconstfvm 5590 f1oiso 5679 mpoeq123 5782 dfoprab4f 6043 fmpox 6050 nnmword 6366 elfi 6809 ltmpig 7089 mul0eqap 8338 qreccl 9330 0fz1 9712 zmodid2 10012 divgcdcoprm0 11622 cnptoprest 12244 txrest 12281 cbvrald 12678 |
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