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Mirrors > Home > ILE Home > Th. List > 3adant1l | GIF version |
Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) |
Ref | Expression |
---|---|
3adant1l.1 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Ref | Expression |
---|---|
3adant1l | ⊢ (((𝜏 ∧ 𝜑) ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3adant1l.1 | . . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | |
2 | 1 | 3expb 1193 | . . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) |
3 | 2 | adantll 468 | . 2 ⊢ (((𝜏 ∧ 𝜑) ∧ (𝜓 ∧ 𝜒)) → 𝜃) |
4 | 3 | 3impb 1188 | 1 ⊢ (((𝜏 ∧ 𝜑) ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-3an 969 |
This theorem is referenced by: 3adant2l 1221 3adant3l 1223 tfrcl 6323 addassnqg 7314 mulassnqg 7316 addasssrg 7688 axaddass 7804 |
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