Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  3adantll3 Structured version   Visualization version   GIF version

Theorem 3adantll3 39023
Description: Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
3adantll3.1 ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
Assertion
Ref Expression
3adantll3 ((((𝜑𝜓𝜂) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Proof of Theorem 3adantll3
StepHypRef Expression
1 simpll1 1098 . . 3 ((((𝜑𝜓𝜂) ∧ 𝜒) ∧ 𝜃) → 𝜑)
2 simpll2 1099 . . 3 ((((𝜑𝜓𝜂) ∧ 𝜒) ∧ 𝜃) → 𝜓)
31, 2jca 554 . 2 ((((𝜑𝜓𝜂) ∧ 𝜒) ∧ 𝜃) → (𝜑𝜓))
4 simplr 791 . 2 ((((𝜑𝜓𝜂) ∧ 𝜒) ∧ 𝜃) → 𝜒)
5 simpr 477 . 2 ((((𝜑𝜓𝜂) ∧ 𝜒) ∧ 𝜃) → 𝜃)
6 3adantll3.1 . 2 ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
73, 4, 5, 6syl21anc 1323 1 ((((𝜑𝜓𝜂) ∧ 𝜒) ∧ 𝜃) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036
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 197  df-an 386  df-3an 1038
This theorem is referenced by:  fourierdlem42  40129  fourierdlem49  40135
  Copyright terms: Public domain W3C validator