Theorem xchbinx 334

Description: Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)

Hypotheses

Ref	Expression
xchbinx.1	⊢ (𝜑 ↔ ¬ 𝜓)
xchbinx.2	⊢ (𝜓 ↔ 𝜒)

Assertion

Ref	Expression
xchbinx	⊢ (𝜑 ↔ ¬ 𝜒)

Proof of Theorem xchbinx

Step	Hyp	Ref	Expression
1		xchbinx.1	. 2 ⊢ (𝜑 ↔ ¬ 𝜓)
2		xchbinx.2	. . 3 ⊢ (𝜓 ↔ 𝜒)
3	2	notbii 320	. 2 ⊢ (¬ 𝜓 ↔ ¬ 𝜒)
4	1, 3	bitri 275	1 ⊢ (𝜑 ↔ ¬ 𝜒)

Syntax hints:  ¬ wn 3   ↔ wb 206

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 207

This theorem is referenced by:  xchbinxr  335  con1bii  356  anor  984  pm4.52  986  pm4.54  988  xordi  1018  xorcom  1514  xorneg1  1522  xorbi12i  1524  norcom  1530  nornot  1531  noran  1532  trunanfal  1582  truxortru  1585  truxorfal  1586  falxorfal  1588  trunortru  1589  trunorfal  1590  falnorfal  1592  nic-mpALT  1672  nic-axALT  1674  sbex  2281  necon3abii  2971  ne3anior  3019  rexab  3655  inssdif0  4325  falseral0  4467  dtruALT  5327  dm0rn0  5867  brprcneu  6812  brprcneuALT  6813  soseq  8092  0nelfz1  13446  pmltpc  25349  cofcutr  27837  nbgrnself  29304  rgrx0ndm  29539  clwwlkneq0  29973  nfrgr2v  30216  frgrncvvdeqlem1  30243  cvbr2  32227  bnj1143  34757  fmlan0  35368  brsset  35867  brtxpsd  35872  dffun10  35892  dfint3  35930  brub  35932  wl-nfeqfb  37514  sbcni  38095  brvdif2  38241  dfssr2  38480  lcvbr2  39005  atlrelat1  39304  dfxor5  43744  df3an2  43746  clsk1independent  44023  spr0nelg  47464  341fppr2  47722  9fppr8  47725  pgrpgt2nabl  48354  lmod1zrnlvec  48483  aacllem  49790