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Metamath Proof Explorer

Theorem 3bitr3d

Description: Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996)

	Ref	Expression
Hypotheses	3bitr3d.1	$⊢ φ \to (ψ \leftrightarrow χ)$
	3bitr3d.2	$⊢ φ \to (ψ \leftrightarrow θ)$
	3bitr3d.3	$⊢ φ \to (χ \leftrightarrow τ)$
Assertion	3bitr3d	$⊢ φ \to (θ \leftrightarrow τ)$

Proof

Step	Hyp	Ref	Expression
1		3bitr3d.1	$⊢ φ \to (ψ \leftrightarrow χ)$
2		3bitr3d.2	$⊢ φ \to (ψ \leftrightarrow θ)$
3		3bitr3d.3	$⊢ φ \to (χ \leftrightarrow τ)$
4	2 1	bitr3d	$⊢ φ \to (θ \leftrightarrow χ)$
5	4 3	bitrd	$⊢ φ \to (θ \leftrightarrow τ)$