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Mirrors > Home > ILE Home > Th. List > sylcom | GIF version |
Description: Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004.) (Proof shortened by O'Cat, 2-Feb-2006.) (Proof shortened by Stefan Allan, 23-Feb-2006.) |
Ref | Expression |
---|---|
sylcom.1 | ⊢ (φ → (ψ → χ)) |
sylcom.2 | ⊢ (ψ → (χ → θ)) |
Ref | Expression |
---|---|
sylcom | ⊢ (φ → (ψ → θ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylcom.1 | . 2 ⊢ (φ → (ψ → χ)) | |
2 | sylcom.2 | . . 3 ⊢ (ψ → (χ → θ)) | |
3 | 2 | a2i 11 | . 2 ⊢ ((ψ → χ) → (ψ → θ)) |
4 | 1, 3 | syl 14 | 1 ⊢ (φ → (ψ → θ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 |
This theorem is referenced by: syl5com 26 syl6 29 syli 33 mpbidi 140 con4biddc 753 jaddc 760 con1biddc 769 necon4addc 2269 necon4bddc 2270 necon4ddc 2271 necon1addc 2275 necon1bddc 2276 dmcosseq 4546 iss 4597 funopg 4877 |
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