Theorem imbi12i 238

Description: Join two logical equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
imbi12i.1	⊢ (𝜑 ↔ 𝜓)
imbi12i.2	⊢ (𝜒 ↔ 𝜃)

Assertion

Ref	Expression
imbi12i	⊢ ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))

Proof of Theorem imbi12i

Step	Hyp	Ref	Expression
1		imbi12i.2	. . 3 ⊢ (𝜒 ↔ 𝜃)
2	1	imbi2i 225	. 2 ⊢ ((𝜑 → 𝜒) ↔ (𝜑 → 𝜃))
3		imbi12i.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
4	3	imbi1i 237	. 2 ⊢ ((𝜑 → 𝜃) ↔ (𝜓 → 𝜃))
5	2, 4	bitri 183	1 ⊢ ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))

Syntax hints:   → wi 4   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  stdcn  832  nfbii  1449  sbi2v  1864  sbim  1924  sb8mo  2011  cbvmo  2037  necon4ddc  2378  raleqbii  2445  rmo5  2644  cbvrmo  2651  ss2ab  3160  snsssn  3683  trint  4036  ssextss  4137  ordsoexmid  4472  zfregfr  4483  tfi  4491  peano2  4504  peano5  4507  relop  4684  dmcosseq  4805  cotr  4915  issref  4916  cnvsym  4917  intasym  4918  intirr  4920  codir  4922  qfto  4923  cnvpom  5076  cnvsom  5077  funcnvuni  5187  poxp  6122  infmoti  6908  dfinfre  8707  bezoutlembi  11682  algcvgblem  11719  isprm2  11787  ntreq0  12290  ss1oel2o  13178