Theorem tbt

Description: A wff is equivalent to its equivalence with a truth. (Contributed by NM, 18-Aug-1993) (Proof shortened by Andrew Salmon, 13-May-2011)

	Ref	Expression
Hypothesis	tbt.1	$$⊢\phi$$
Assertion	tbt	$$⊢\psi \leftrightarrow \left(\psi \leftrightarrow \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		tbt.1	$$⊢\phi$$
2		ibibr	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \left(\psi \leftrightarrow \phi \right)\right)$$
3	2	pm5.74ri	$$⊢\phi \to \left(\psi \leftrightarrow \left(\psi \leftrightarrow \phi \right)\right)$$
4	1 3	ax-mp	$$⊢\psi \leftrightarrow \left(\psi \leftrightarrow \phi \right)$$