Theorem 3bitr2ri 265

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 243	. 2 ⊢ (φ ↔ χ)
4		3bitr2i.3	. 2 ⊢ (χ ↔ θ)
5	3, 4	bitr2i 241	1 ⊢ (θ ↔ φ)

Syntax hints:   ↔ wb 176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  sbnf2  2108  ssrab  3345  unipw1  4326  dmopab3  4918  dfres2  5003  ssrnres  5060  df2nd2  5112  dfnnc3  5886  enex  6032