Theorem adantrrl 737

Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)

Hypothesis

Ref	Expression
adantr2.1	⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)

Assertion

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

Proof of Theorem adantrrl

Step	Hyp	Ref	Expression
1		simpr 490	. 2 ⊢ ((𝜏 ∧ 𝜒) → 𝜒)
2		adantr2.1	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)
3	1, 2	sylanr2 696	1 ⊢ ((𝜑 ∧ (𝜓 ∧ (𝜏 ∧ 𝜒))) → 𝜃)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  zorn2lem6  10503  ltmul12a  12089  mndind  18918  neiint  23298  neissex  23321  1stcfb  23639  1stcrest  23647  grporcan  30907  mdslmd3i  32721  colineardim1  36574  cvratlem  40236  ps-2  40293  fsuppssind  43366