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Mirrors > Home > ILE Home > Th. List > ancomsd | GIF version |
Description: Deduction commuting conjunction in antecedent. (Contributed by NM, 12-Dec-2004.) |
Ref | Expression |
---|---|
ancomsd.1 | ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) |
Ref | Expression |
---|---|
ancomsd | ⊢ (𝜑 → ((𝜒 ∧ 𝜓) → 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 253 | . 2 ⊢ ((𝜒 ∧ 𝜓) ↔ (𝜓 ∧ 𝜒)) | |
2 | ancomsd.1 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) | |
3 | 1, 2 | syl5bi 141 | 1 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) → 𝜃)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: sylan2d 278 mpand 405 anabsi6 514 ralxfrd 4194 rexxfrd 4195 poirr2 4717 smoel 5915 genprndl 6619 genprndu 6620 addcanprlemu 6713 leltadd 7442 lemul12b 7827 lbzbi 8551 |
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