Theorem anidm 400

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		pm4.24 399	. 2 ⊢ (𝜑 ↔ (𝜑 ∧ 𝜑))
2	1	bicomi 132	1 ⊢ ((𝜑 ∧ 𝜑) ↔ 𝜑)

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  anidmdbi  402  anandi  598  anandir  599  truantru  1450  falanfal  1453  truxortru  1468  truxorfal  1469  falxortru  1470  falxorfal  1471  sbnf2  2041  2eu4  2180  inidm  3440  ralidm  3628  opcom  4393  opeqsn  4395  poirr  4454  rnxpid  5225  xp11m  5229  fununi  5452  brprcneu  5691  erinxp  6887  dom2lem  7062  dmaddpi  7698  dmmulpi  7699  enq0ref  7806  enq0tr  7807  msqap0  9004  expap0  11027  sqap0  11064  xrmaxiflemcom  12040  gcddvds  12765  isnsg2  14065  eqger  14086  xmeter  15662  clwwlkn2  16867  2alsraln0idm  17366