Theorem bicomdd

Description: Commute two sides of a biconditional in a deduction. (Contributed by Rodolfo Medina, 19-Oct-2010) (Proof shortened by Andrew Salmon, 29-Jun-2011)

	Ref	Expression
Hypothesis	bicomdd.1	$$⊢\phi \to \left(\psi \to \left(\chi \leftrightarrow \theta \right)\right)$$
Assertion	bicomdd	$$⊢\phi \to \left(\psi \to \left(\theta \leftrightarrow \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bicomdd.1	$$⊢\phi \to \left(\psi \to \left(\chi \leftrightarrow \theta \right)\right)$$
2		bicom	$$⊢\left(\chi \leftrightarrow \theta \right)\leftrightarrow \left(\theta \leftrightarrow \chi \right)$$
3	1 2	imbitrdi	$$⊢\phi \to \left(\psi \to \left(\theta \leftrightarrow \chi \right)\right)$$