Theorem 3bitr3ri 267

Description: A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
3bitr3i.1	⊢ (φ ↔ ψ)
3bitr3i.2	⊢ (φ ↔ χ)
3bitr3i.3	⊢ (ψ ↔ θ)

Assertion

Ref	Expression
3bitr3ri	⊢ (θ ↔ χ)

Proof of Theorem 3bitr3ri

Step	Hyp	Ref	Expression
1		3bitr3i.3	. 2 ⊢ (ψ ↔ θ)
2		3bitr3i.1	. . 3 ⊢ (φ ↔ ψ)
3		3bitr3i.2	. . 3 ⊢ (φ ↔ χ)
4	2, 3	bitr3i 242	. 2 ⊢ (ψ ↔ χ)
5	1, 4	bitr3i 242	1 ⊢ (θ ↔ χ)

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:  bigolden  901  2eu8  2291  2ralor  2781  sbcco  3069  dfiin2g  4001  nnadjoinpw  4522  el1st  4730  dffun6f  5124  fununi  5161