Theorem 3adantl3 1157

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
3adantl3	⊢ (((𝜑 ∧ 𝜓 ∧ 𝜏) ∧ 𝜒) → 𝜃)

Proof of Theorem 3adantl3

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

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980

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 982

This theorem is referenced by:  ltsopr  7663  lediv2a  8922  muldvds1  11981  muldvds2  11982  dvdscmul  11983  dvdsmulc  11984  rpexp  12321  iscnp4  14454  cnpnei  14455  xblm  14653