Theorem imbi12i 239

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 226	. 2 ⊢ ((𝜑 → 𝜒) ↔ (𝜑 → 𝜃))
3		imbi12i.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
4	3	imbi1i 238	. 2 ⊢ ((𝜑 → 𝜃) ↔ (𝜓 → 𝜃))
5	2, 4	bitri 184	1 ⊢ ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))

This proof depends on syntax axioms:   → wi 4   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  stdcn  859  dcfromcon  1498  dcfrompeirce  1499  nfbii  1526  sbi2v  1947  sbim  2013  sb8mo  2100  cbvmo  2126  necon4ddc  2492  raleqbii  2562  rmo5  2773  cbvrmo  2785  ss2ab  3316  snsssn  3886  trint  4246  ssextss  4362  ordsoexmid  4711  zfregfr  4723  tfi  4731  peano2  4744  peano5  4747  relop  4932  dmcosseq  5056  cotr  5172  issref  5173  cnvsym  5174  intasym  5175  intirr  5177  codir  5179  qfto  5180  cnvpom  5333  cnvsom  5334  funcnvuni  5453  poxp  6472  infmoti  7374  dfinfre  9294  bezoutlembi  12807  algcvgblem  12852  isprm2  12920  ntreq0  15358  ss1oel2o  17222  alsbii  17348  ralsbii  17349  alseubii  17380  ralseubii  17381