Theorem tbt 245

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

Hypothesis

Ref	Expression
tbt.1	⊢ 𝜑

Assertion

Ref	Expression
tbt	⊢ (𝜓 ↔ (𝜓 ↔ 𝜑))

Proof of Theorem tbt

Step	Hyp	Ref	Expression
1		tbt.1	. 2 ⊢ 𝜑
2		ibibr 244	. . 3 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → (𝜓 ↔ 𝜑)))
3	2	pm5.74ri 179	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ↔ 𝜑)))
4	1, 3	ax-mp 7	1 ⊢ (𝜓 ↔ (𝜓 ↔ 𝜑))

Syntax hints:   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  tbtru  1294  exists1  2037  reu6  2781  eqv  3267  vprc  3909  bj-vprc  10687