Theorem tbt 372

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 371	. . 3 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → (𝜓 ↔ 𝜑)))
3	2	pm5.74ri 275	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ↔ 𝜑)))
4	1, 3	ax-mp 5	1 ⊢ (𝜓 ↔ (𝜓 ↔ 𝜑))

This proof depends on syntax axioms:   ↔ wb 209

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

This proof depends on definitions:  df-bi 210

This theorem is used by:  tbtru  1578  eqv  3468  eqvf  3469  abv  3470  pm13.183  3628  reu6  3692  ab0orv  4342  vnexOLD  5286  iotanul  6523  eqelbid  32858  elnev  45187