This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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	$⊢ φ$
	Assertion	tbt	$⊢ ψ \leftrightarrow (ψ \leftrightarrow φ)$

Proof

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