Theorem anidm 396

Description: Idempotent law for conjunction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Mar-2014.)

Assertion

Ref	Expression
anidm	⊢ ((𝜑 ∧ 𝜑) ↔ 𝜑)

Proof of Theorem anidm

Step	Hyp	Ref	Expression
1		pm4.24 395	. 2 ⊢ (𝜑 ↔ (𝜑 ∧ 𝜑))
2	1	bicomi 132	1 ⊢ ((𝜑 ∧ 𝜑) ↔ 𝜑)

Syntax hints:   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  anidmdbi  398  anandi  590  anandir  591  truantru  1401  falanfal  1404  truxortru  1419  truxorfal  1420  falxortru  1421  falxorfal  1422  sbnf2  1981  2eu4  2119  inidm  3344  ralidm  3523  opcom  4250  opeqsn  4252  poirr  4307  rnxpid  5063  xp11m  5067  fununi  5284  brprcneu  5508  erinxp  6608  dom2lem  6771  dmaddpi  7323  dmmulpi  7324  enq0ref  7431  enq0tr  7432  expap0  10547  sqap0  10583  xrmaxiflemcom  11252  gcddvds  11958  isnsg2  13016  eqger  13036  xmeter  13829