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  4320  uniuni  4542  rabxp  4756  eliunxp  4861  dmmrnm  4943  imadisj  5090  intirr  5115  resco  5233  funcnv3  5383  fncnv  5387  fun11  5388  fununi  5389  f1mpt  5901  mpomptx  6101  ixp0x  6881  mapsnen  6972  xpcomco  6993  enq0tr  7629  elq  9825  bitsmod  12475  pythagtrip  12814  ntreq0  14814  tx1cn  14951