Theorem 3adantl2 1106

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
3adantl2	⊢ (((𝜑 ∧ 𝜏 ∧ 𝜓) ∧ 𝜒) → 𝜃)

Proof of Theorem 3adantl2

Step	Hyp	Ref	Expression
1		3simpb 947	. 2 ⊢ ((𝜑 ∧ 𝜏 ∧ 𝜓) → (𝜑 ∧ 𝜓))
2		3adantl.1	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
3	1, 2	sylan 279	1 ⊢ (((𝜑 ∧ 𝜏 ∧ 𝜓) ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 930

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 932

This theorem is referenced by:  3ad2antl1  1111  nnmord  6343  ltaprg  7328  lediv2a  8511  zdiv  8991  neiint  12096  cnpnei  12169