Theorem biantrurd 300

Description: A wff is equivalent to its conjunction with truth. (Contributed by NM, 1-May-1995.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Hypothesis

Ref	Expression
biantrud.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
biantrurd	⊢ (𝜑 → (𝜒 ↔ (𝜓 ∧ 𝜒)))

Proof of Theorem biantrurd

Step	Hyp	Ref	Expression
1		biantrud.1	. 2 ⊢ (𝜑 → 𝜓)
2		ibar 296	. 2 ⊢ (𝜓 → (𝜒 ↔ (𝜓 ∧ 𝜒)))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝜒 ↔ (𝜓 ∧ 𝜒)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104

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

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  3anibar  1112  drex1  1727  elrab3t  2771  bnd2  4014  opbrop  4530  opelresi  4737  funcnv3  5089  fnssresb  5139  dff1o5  5275  fneqeql2  5422  fnniniseg2  5436  rexsupp  5437  dffo3  5460  fmptco  5478  fnressn  5497  fconst4m  5531  riota2df  5642  eloprabga  5749  frecabcl  6178  mptelixpg  6505  exmidfodomrlemreseldju  6887  enq0breq  7056  genpassl  7144  genpassu  7145  elnnnn0  8777  peano2z  8847  znnsub  8862  znn0sub  8876  uzin  9112  nn01to3  9163  rpnegap  9227  divelunit  9480  elfz5  9493  uzsplit  9567  elfzonelfzo  9702  adddivflid  9760  divfl0  9764  2shfti  10326  rexuz3  10484  clim2c  10733  fisumss  10845  infssuzex  11284  bezoutlemmain  11326  eltg3  11818  mulc1cncf  11918