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	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
bibi2i	⊢ ((𝜒 ↔ 𝜑) ↔ (𝜒 ↔ 𝜓))

Proof of Theorem bibi2i

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ ((𝜒 ↔ 𝜑) → (𝜒 ↔ 𝜑))
2		bibi2i.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	1, 2	bitrdi 287	. 2 ⊢ ((𝜒 ↔ 𝜑) → (𝜒 ↔ 𝜓))
4		id 22	. . 3 ⊢ ((𝜒 ↔ 𝜓) → (𝜒 ↔ 𝜓))
5	4, 2	bitr4di 289	. 2 ⊢ ((𝜒 ↔ 𝜓) → (𝜒 ↔ 𝜑))
6	3, 5	impbii 208	1 ⊢ ((𝜒 ↔ 𝜑) ↔ (𝜒 ↔ 𝜓))

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  33732  fineqvrep  34095  axrepprim  34671  brtxpsd3  34868  bisym1  35304  bj-bixor  35469  eliminable-veqab  35745  bj-snsetex  35844  bj-reabeq  35908  bj-clex  35912  wl-3xorbi  36354  sn-axrep5v  41033  ifpidg  42242  nanorxor  43064  mo0sn  47500