Theorem 3bitr2ri 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
3bitr2ri	⊢ (𝜃 ↔ 𝜑)

Proof of Theorem 3bitr2ri

Step	Hyp	Ref	Expression
1		3bitr2i.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2		3bitr2i.2	. . 3 ⊢ (𝜒 ↔ 𝜓)
3	1, 2	bitr4i 186	. 2 ⊢ (𝜑 ↔ 𝜒)
4		3bitr2i.3	. 2 ⊢ (𝜒 ↔ 𝜃)
5	3, 4	bitr2i 184	1 ⊢ (𝜃 ↔ 𝜑)

Syntax hints:   ↔ wb 104

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  sbnf2  1930  ssrab  3139  rabn0m  3354  unidif0  4049  relop  4647  dmopab3  4710  issref  4877  fununi  5147  cnvoprab  6082  ssfirab  6771