Theorem tbt 369

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.)

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 368	. . 3 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → (𝜓 ↔ 𝜑)))
3	2	pm5.74ri 272	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ↔ 𝜑)))
4	1, 3	ax-mp 5	1 ⊢ (𝜓 ↔ (𝜓 ↔ 𝜑))

Syntax hints:   ↔ wb 206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207

This theorem is referenced by:  tbtru  1548  eqv  3454  eqvf  3455  abv  3456  pm13.183  3629  reu6  3694  vnex  5264  iotanul  6477  eqelbid  32454  elnev  44420