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  1936  sbexyz  2054  exists1  2174  euxfrdc  2989  euind  2990  rmo4  2996  rmo3f  3000  rmo3  3121  ddifstab  3336  opm  4319  uniuni  4539  rabxp  4753  eliunxp  4858  dmmrnm  4939  imadisj  5086  intirr  5111  resco  5229  funcnv3  5379  fncnv  5383  fun11  5384  fununi  5385  f1mpt  5888  mpomptx  6086  ixp0x  6863  mapsnen  6954  xpcomco  6973  enq0tr  7609  elq  9805  bitsmod  12453  pythagtrip  12792  ntreq0  14791  tx1cn  14928