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Mirrors > Home > ILE Home > Th. List > 3adantl3 | GIF version |
Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.) |
Ref | Expression |
---|---|
3adantl.1 | ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) |
Ref | Expression |
---|---|
3adantl3 | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜏) ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpa 935 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜏) → (𝜑 ∧ 𝜓)) | |
2 | 3adantl.1 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) | |
3 | 1, 2 | sylan 277 | 1 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜏) ∧ 𝜒) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 919 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 |
This theorem depends on definitions: df-bi 115 df-3an 921 |
This theorem is referenced by: ltsopr 6786 lediv2a 7973 expival 9478 muldvds1 10220 muldvds2 10221 dvdscmul 10222 dvdsmulc 10223 rpexp 10532 |
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