Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  3adantll2 Structured version   Visualization version   GIF version

 Description: Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
3adantll2.1 ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
Assertion
Ref Expression
3adantll2 ((((𝜑𝜂𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Proof of Theorem 3adantll2
StepHypRef Expression
1 simpll1 1093 . . 3 ((((𝜑𝜂𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜑)
2 simpll3 1095 . . 3 ((((𝜑𝜂𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜓)
31, 2jca 553 . 2 ((((𝜑𝜂𝜓) ∧ 𝜒) ∧ 𝜃) → (𝜑𝜓))
4 simplr 788 . 2 ((((𝜑𝜂𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜒)
5 simpr 476 . 2 ((((𝜑𝜂𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜃)
6 3adantll2.1 . 2 ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
73, 4, 5, 6syl21anc 1317 1 ((((𝜑𝜂𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031 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 196  df-an 385  df-3an 1033 This theorem is referenced by:  icccncfext  38773  fourierdlem42  39042
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