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Mirrors > Home > ILE Home > Th. List > ancom2s | GIF version |
Description: Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) |
Ref | Expression |
---|---|
an12s.1 | ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) |
Ref | Expression |
---|---|
ancom2s | ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜓)) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.22 265 | . 2 ⊢ ((𝜒 ∧ 𝜓) → (𝜓 ∧ 𝜒)) | |
2 | an12s.1 | . 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) | |
3 | 1, 2 | sylan2 286 | 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 is referenced by: an42s 589 ordsuc 4580 xpexr2m 5088 f1elima 5795 f1imaeq 5797 isosolem 5846 caovlem2d 6090 2ndconst 6248 isotilem 7036 prarloclem4 7528 mulsub 8389 leltadd 8435 eqord1 8471 divmul24ap 8704 fprodseq 11626 grpidpropdg 12853 cmnpropd 13251 unitpropdg 13515 blcomps 14373 blcom 14374 cxple 14814 cxple3 14818 |
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