Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ad4ant134 Structured version   Visualization version   GIF version

 Description: Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
Hypothesis
Ref Expression
ad4ant3.1 ((𝜑𝜓𝜒) → 𝜃)
Assertion
Ref Expression
ad4ant134 ((((𝜑𝜏) ∧ 𝜓) ∧ 𝜒) → 𝜃)

Proof of Theorem ad4ant134
StepHypRef Expression
1 ad4ant3.1 . . 3 ((𝜑𝜓𝜒) → 𝜃)
213expa 1115 . 2 (((𝜑𝜓) ∧ 𝜒) → 𝜃)
32adantllr 718 1 ((((𝜑𝜏) ∧ 𝜓) ∧ 𝜒) → 𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084 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 210  df-an 400  df-3an 1086 This theorem is referenced by:  ad5ant245  1358  ad5ant134  1364  ad5ant135  1365  ad5ant145  1366  ralxfrd2  5296  gruwun  10222  lemul12b  11484  initoeu1  17262  termoeu1  17269  quscrng  20001  metss  23106  wlkswwlksf1o  27656  climxlim2lem  42344  smflimlem4  43264
 Copyright terms: Public domain W3C validator