Theorem tbt 333

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 332	. . 3 ⊢ ((φ → ψ) ↔ (φ → (ψ ↔ φ)))
3	2	pm5.74ri 237	. 2 ⊢ (φ → (ψ ↔ (ψ ↔ φ)))
4	1, 3	ax-mp 5	1 ⊢ (ψ ↔ (ψ ↔ φ))

This proof depends on syntax axioms:   ↔ wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177

This theorem is used by:  tbtru  1324  exists1  2293  reu6  3026  eqv  3566