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  3345  ralidm  3524  opcom  4251  opeqsn  4253  poirr  4308  rnxpid  5064  xp11m  5068  fununi  5285  brprcneu  5509  erinxp  6609  dom2lem  6772  dmaddpi  7324  dmmulpi  7325  enq0ref  7432  enq0tr  7433  expap0  10550  sqap0  10587  xrmaxiflemcom  11257  gcddvds  11964  isnsg2  13063  eqger  13083  xmeter  13939