MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ad5ant2345 Structured version   Visualization version   GIF version

Theorem ad5ant2345 1436
Description: Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
Hypothesis
Ref Expression
ad5ant2345.1 ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
Assertion
Ref Expression
ad5ant2345 (((((𝜂𝜑) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Proof of Theorem ad5ant2345
StepHypRef Expression
1 ad5ant2345.1 . . . 4 ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
21exp41 639 . . 3 (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
32adantl 473 . 2 ((𝜂𝜑) → (𝜓 → (𝜒 → (𝜃𝜏))))
43imp41 620 1 (((((𝜂𝜑) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383
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 385
This theorem is referenced by:  mblfinlem2  33679  liminflelimsuplem  40427  climxlim2lem  40491  iundjiun  41097
  Copyright terms: Public domain W3C validator