Theorem 3adantl1 1153

Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)

Hypothesis

Ref	Expression
3adantl.1	⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
3adantl1	⊢ (((𝜏 ∧ 𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)

Proof of Theorem 3adantl1

Step	Hyp	Ref	Expression
1		3simpc 996	. 2 ⊢ ((𝜏 ∧ 𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜓))
2		3adantl.1	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
3	1, 2	sylan 283	1 ⊢ (((𝜏 ∧ 𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 978

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  df-3an 980

This theorem is referenced by:  3ad2antl2  1160  3ad2antl3  1161  distrlem1prl  7584  distrlem1pru  7585  divmuldivap  8672  modqaddmulmod  10394  expnlbnd  10648  lcmledvds  12073  ctinf  12434  upxp  13912