Theorem 3bitrri 207

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

Hypotheses

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

Assertion

Ref	Expression
3bitrri	⊢ (𝜃 ↔ 𝜑)

Proof of Theorem 3bitrri

Step	Hyp	Ref	Expression
1		3bitri.3	. 2 ⊢ (𝜒 ↔ 𝜃)
2		3bitri.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
3		3bitri.2	. . 3 ⊢ (𝜓 ↔ 𝜒)
4	2, 3	bitr2i 185	. 2 ⊢ (𝜒 ↔ 𝜑)
5	1, 4	bitr3i 186	1 ⊢ (𝜃 ↔ 𝜑)

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:  sbralie  2744  reu8  2956  unass  3316  ssin  3381  difab  3428  iunss  3953  poirr  4338  cnvuni  4848  dfco2  5165  dff1o6  5819  elznn0  9332  bj-ssom  15428