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Metamath Proof Explorer

Theorem 3bitr4i

Description: A chained inference from transitive law for logical equivalence. This inference is frequently used to apply a definition to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993)

	Ref	Expression
Hypotheses	3bitr4i.1	$⊢ φ \leftrightarrow ψ$
	3bitr4i.2	$⊢ χ \leftrightarrow φ$
	3bitr4i.3	$⊢ θ \leftrightarrow ψ$
Assertion	3bitr4i	$⊢ χ \leftrightarrow θ$

Proof

Step	Hyp	Ref	Expression
1		3bitr4i.1	$⊢ φ \leftrightarrow ψ$
2		3bitr4i.2	$⊢ χ \leftrightarrow φ$
3		3bitr4i.3	$⊢ θ \leftrightarrow ψ$
4	1 3	bitr4i	$⊢ φ \leftrightarrow θ$
5	2 4	bitri	$⊢ χ \leftrightarrow θ$