MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  anabs5 Structured version   Visualization version   GIF version

Theorem anabs5 653
Description: Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
Assertion
Ref Expression
anabs5 ((𝜑 ∧ (𝜑𝜓)) ↔ (𝜑𝜓))

Proof of Theorem anabs5
StepHypRef Expression
1 ibar 524 . . 3 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
21bicomd 214 . 2 (𝜑 → ((𝜑𝜓) ↔ 𝜓))
32pm5.32i 570 1 ((𝜑 ∧ (𝜑𝜓)) ↔ (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384
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 198  df-an 385
This theorem is referenced by:  rexanid  3065  reuanid  3066  rmoanid  3067  axrep5  4935  axsep2  4941  bj-axrep5  33154  elinintrab  38490  2sb5nd  39370  eelTT1  39529  uun121  39603  uunTT1  39613  uunTT1p1  39614  uunTT1p2  39615  uun111  39625  uun2221  39633  uun2221p1  39634  uun2221p2  39635  2sb5ndVD  39730  2sb5ndALT  39752
  Copyright terms: Public domain W3C validator