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 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:  wfrlem12OLD  8360  ecopovtrn  8860  rrxmet  25442  nvaddsub4  30676  adjlnop  32105  pl1cn  33954  rrnmet  37836  lflsub  39068  lflmul  39069  cvlatexch3  39339  cdleme5  40242  cdlemeg46rjgN  40524  cdlemg2l  40605  cdlemg10c  40641  tendospcanN  41025  dicvaddcl  41192  dicvscacl  41193  dochexmidlem8  41469  limsupre3lem  45747  fourierdlem42  46164  fourierdlem113  46234  ovnsupge0  46572  ovncvrrp  46579  ovnhoilem2  46617