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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 264 | . 2 ⊢ ((𝜒 ∧ 𝜓) ↔ (𝜓 ∧ 𝜒)) | |
2 | ancomsd.1 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) | |
3 | 1, 2 | syl5bi 151 | 1 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) → 𝜃)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 |
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: sylan2d 290 mpand 423 anabsi6 550 ralxfrd 4321 rexxfrd 4322 poirr2 4867 smoel 6127 genprndl 7230 genprndu 7231 addcanprlemu 7324 leltadd 8076 lemul12b 8477 lbzbi 9258 dvdssub2 11330 |
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