Theorem 3ad2antr1 1193

Description: Deduction adding a conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.)

Hypothesis

Ref	Expression
3ad2antl.1	⊢ ((𝜑 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
3ad2antr1	⊢ ((𝜑 ∧ (𝜒 ∧ 𝜓 ∧ 𝜏)) → 𝜃)

Proof of Theorem 3ad2antr1

Step	Hyp	Ref	Expression
1		3ad2antl.1	. . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
2	1	adantrr 483	. 2 ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜓)) → 𝜃)
3	2	3adantr3 1189	1 ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜓 ∧ 𝜏)) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1009

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 1011

This theorem is referenced by:  ispod  4444  poxp  6458  fzosubel2  10591  hashdifpr  11239  pfxccat3a  11488  grpsubadd  13870  mulgnnass  13937  mulgnn0ass  13938  issubg2m  13969  srgdilem  14247  lsssn0  14679  dvconst  15718  dvconstre  15720  isclwwlk  16549  clwwlkccatlem  16555  clwwlkccat  16556