MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  anabss1 Structured version   Visualization version   GIF version

Theorem anabss1 664
Description: Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
Hypothesis
Ref Expression
anabss1.1 (((𝜑𝜓) ∧ 𝜑) → 𝜒)
Assertion
Ref Expression
anabss1 ((𝜑𝜓) → 𝜒)

Proof of Theorem anabss1
StepHypRef Expression
1 anabss1.1 . . 3 (((𝜑𝜓) ∧ 𝜑) → 𝜒)
21an32s 650 . 2 (((𝜑𝜑) ∧ 𝜓) → 𝜒)
32anabsan 663 1 ((𝜑𝜓) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398
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 209  df-an 399
This theorem is referenced by:  anabss4  665  ordtri3or  6195  onfununi  7952  omordi  8166  oeoelem  8198  hashssdif  13754  fzindd  39130  nzss  40800  stirlinglem5  42507
  Copyright terms: Public domain W3C validator