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  1412  falanfal  1415  truxortru  1430  truxorfal  1431  falxortru  1432  falxorfal  1433  sbnf2  2000  2eu4  2138  inidm  3373  ralidm  3552  opcom  4284  opeqsn  4286  poirr  4343  rnxpid  5105  xp11m  5109  fununi  5327  brprcneu  5554  erinxp  6677  dom2lem  6840  dmaddpi  7409  dmmulpi  7410  enq0ref  7517  enq0tr  7518  expap0  10678  sqap0  10715  xrmaxiflemcom  11431  gcddvds  12155  isnsg2  13409  eqger  13430  xmeter  14756