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 271	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ↔ 𝜑)))
4	1, 3	ax-mp 5	1 ⊢ (𝜓 ↔ (𝜓 ↔ 𝜑))

Syntax hints:   ↔ wb 205

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 206

This theorem is referenced by:  tbtru  1549  eqv  3439  eqvf  3440  abv  3441  reu6  3664  vnex  5241  iotanul  6408  elnev  42009