Theorem biantrud 304

Description: A wff is equivalent to its conjunction with truth. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2013.)

Hypothesis

Ref	Expression
biantrud.1	⊢ (𝜑 → 𝜓)

Assertion

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

Proof of Theorem biantrud

Step	Hyp	Ref	Expression
1		biantrud.1	. 2 ⊢ (𝜑 → 𝜓)
2		iba 300	. 2 ⊢ (𝜓 → (𝜒 ↔ (𝜒 ∧ 𝜓)))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝜒 ↔ (𝜒 ∧ 𝜓)))

Syntax hints:   → wi 4   ∧ 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:  mpbiran2d  442  posng  4700  elrnmpt1  4880  fliftf  5802  elxp7  6173  eroveu  6628  sbthlemi5  6962  sbthlemi6  6963  elfi2  6973  reapltxor  8548  divap0b  8642  nnle1eq1  8945  nn0le0eq0  9206  nn0lt10b  9335  ioopos  9952  xrmaxiflemcom  11259  fz1f1o  11385  nndivdvds  11805  dvdsmultr2  11842  pcmpt  12343  pcmpt2  12344  lssle0  13463  discld  13721  cncnpi  13813  cnptoprest2  13825  lmss  13831  txcn  13860  isxmet2d  13933  xblss2  13990  bdxmet  14086  xmetxp  14092  cncfcdm  14154  lgsneg  14510  lgsdilem  14513