Theorem bibi1d

Description: Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 11-May-1993)

	Ref	Expression
Hypothesis	imbid.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	bibi1d	$$⊢\phi \to \left(\left(\psi \leftrightarrow \theta \right)\leftrightarrow \left(\chi \leftrightarrow \theta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		imbid.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2	1	bibi2d	$$⊢\phi \to \left(\left(\theta \leftrightarrow \psi \right)\leftrightarrow \left(\theta \leftrightarrow \chi \right)\right)$$
3		bicom	$$⊢\left(\psi \leftrightarrow \theta \right)\leftrightarrow \left(\theta \leftrightarrow \psi \right)$$
4		bicom	$$⊢\left(\chi \leftrightarrow \theta \right)\leftrightarrow \left(\theta \leftrightarrow \chi \right)$$
5	2 3 4	3bitr4g	$$⊢\phi \to \left(\left(\psi \leftrightarrow \theta \right)\leftrightarrow \left(\chi \leftrightarrow \theta \right)\right)$$