Theorem 3bitr2i 208

Description: A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)

Hypotheses

Ref	Expression
3bitr2i.1	⊢ (𝜑 ↔ 𝜓)
3bitr2i.2	⊢ (𝜒 ↔ 𝜓)
3bitr2i.3	⊢ (𝜒 ↔ 𝜃)

Assertion

Ref	Expression
3bitr2i	⊢ (𝜑 ↔ 𝜃)

Proof of Theorem 3bitr2i

Step	Hyp	Ref	Expression
1		3bitr2i.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2		3bitr2i.2	. . 3 ⊢ (𝜒 ↔ 𝜓)
3	1, 2	bitr4i 187	. 2 ⊢ (𝜑 ↔ 𝜒)
4		3bitr2i.3	. 2 ⊢ (𝜒 ↔ 𝜃)
5	3, 4	bitri 184	1 ⊢ (𝜑 ↔ 𝜃)

Syntax hints:   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  an13  563  sbanv  1904  sbexyz  2022  exists1  2141  euxfrdc  2950  euind  2951  rmo4  2957  rmo3f  2961  rmo3  3081  ddifstab  3295  opm  4267  uniuni  4486  rabxp  4700  eliunxp  4805  dmmrnm  4885  imadisj  5031  intirr  5056  resco  5174  funcnv3  5320  fncnv  5324  fun11  5325  fununi  5326  f1mpt  5818  mpomptx  6013  ixp0x  6785  mapsnen  6870  xpcomco  6885  enq0tr  7501  elq  9696  pythagtrip  12452  ntreq0  14368  tx1cn  14505