Theorem 3adant3l 1181

Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 25-Jun-2022.)

Hypothesis

Ref	Expression
ad4ant3.1	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
3adant3l	⊢ ((𝜑 ∧ 𝜓 ∧ (𝜏 ∧ 𝜒)) → 𝜃)

Proof of Theorem 3adant3l

Step	Hyp	Ref	Expression
1		simpr 484	. 2 ⊢ ((𝜏 ∧ 𝜒) → 𝜒)
2		ad4ant3.1	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
3	1, 2	syl3an3 1165	1 ⊢ ((𝜑 ∧ 𝜓 ∧ (𝜏 ∧ 𝜒)) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088

This theorem is referenced by:  ecopovtrn  8796  rrxmet  25315  nvaddsub4  30593  adjlnop  32022  pl1cn  33952  rrnmet  37830  lflsub  39067  lflmul  39068  cvlatexch3  39338  cdleme5  40241  cdlemeg46rjgN  40523  cdlemg2l  40604  cdlemg10c  40640  tendospcanN  41024  dicvaddcl  41191  dicvscacl  41192  dochexmidlem8  41468  limsupre3lem  45737  fourierdlem42  46154  fourierdlem113  46224  ovnsupge0  46562  ovncvrrp  46569  ovnhoilem2  46607