MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bibi2i Structured version   Visualization version   GIF version

Theorem bibi2i 338
Description: Inference adding a biconditional to the left in an equivalence. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 16-May-2013.)
Hypothesis
Ref Expression
bibi2i.1 (𝜑𝜓)
Assertion
Ref Expression
bibi2i ((𝜒𝜑) ↔ (𝜒𝜓))

Proof of Theorem bibi2i
StepHypRef Expression
1 id 22 . . 3 ((𝜒𝜑) → (𝜒𝜑))
2 bibi2i.1 . . 3 (𝜑𝜓)
31, 2bitrdi 287 . 2 ((𝜒𝜑) → (𝜒𝜓))
4 id 22 . . 3 ((𝜒𝜓) → (𝜒𝜓))
54, 2bitr4di 289 . 2 ((𝜒𝜓) → (𝜒𝜑))
63, 5impbii 208 1 ((𝜒𝜑) ↔ (𝜒𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 206
This theorem is referenced by:  bibi1i  339  bibi12i  340  bibi2d  343  con2bi  354  pm4.71r  560  xorass  1515  sblbis  2306  sbrbif  2308  eqabbw  2810  eqabbOLD  2875  eqabf  2936  pm13.183  3657  dfss2  3969  eq0  4344  eq0rdv  4405  disj  4448  disj3  4454  rzal  4509  axrep5  5292  axrep6  5293  axsepgfromrep  5298  ax6vsep  5304  inex1  5318  axprALT  5421  zfpair2  5429  sucel  6439  suppvalbr  8150  bnj89  33763  fineqvrep  34126  axrepprim  34702  brtxpsd3  34899  bisym1  35352  bj-bixor  35517  eliminable-veqab  35793  bj-snsetex  35892  bj-reabeq  35956  bj-clex  35960  wl-3xorbi  36402  sn-axrep5v  41081  ifpidg  42290  nanorxor  43112  mo0sn  47548
  Copyright terms: Public domain W3C validator