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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 ((𝜑𝜓𝜒) → 𝜃)
Assertion
Ref Expression
3adant3l ((𝜑𝜓 ∧ (𝜏𝜒)) → 𝜃)

Proof of Theorem 3adant3l
StepHypRef Expression
1 simpr 484 . 2 ((𝜏𝜒) → 𝜒)
2 ad4ant3.1 . 2 ((𝜑𝜓𝜒) → 𝜃)
31, 2syl3an3 1166 1 ((𝜑𝜓 ∧ (𝜏𝜒)) → 𝜃)
Colors of variables: wff setvar class
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
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