Theorem 3adant3l 1182

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 1166	1 ⊢ ((𝜑 ∧ 𝜓 ∧ (𝜏 ∧ 𝜒)) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087

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 1089

This theorem is referenced by:  ecopovtrn  8771  rrxmet  25381  nvaddsub4  30751  adjlnop  32180  pl1cn  34139  rrnmet  38109  lflsub  39472  lflmul  39473  cvlatexch3  39743  cdleme5  40645  cdlemeg46rjgN  40927  cdlemg2l  41008  cdlemg10c  41044  tendospcanN  41428  dicvaddcl  41595  dicvscacl  41596  dochexmidlem8  41872  limsupre3lem  46119  fourierdlem42  46536  fourierdlem113  46606  ovnsupge0  46944  ovncvrrp  46951  ovnhoilem2  46989