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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 266 | . 2 ⊢ ((𝜒 ∧ 𝜓) ↔ (𝜓 ∧ 𝜒)) | |
2 | ancomsd.1 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) | |
3 | 1, 2 | biimtrid 152 | 1 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) → 𝜃)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: sylan2d 294 mpand 429 anabsi6 580 ralxfrd 4462 rexxfrd 4463 poirr2 5021 smoel 6300 genprndl 7519 genprndu 7520 addcanprlemu 7613 leltadd 8403 lemul12b 8817 lbzbi 9615 dvdssub2 11841 odzdvds 12244 |
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