Theorem monothetic 266

Description: Two self-implications (see id 22) are equivalent. This theorem, rather trivial in our axiomatization, is (the biconditional form of) a standard axiom for monothetic BCI logic. This is the most general theorem of which trujust 1539 is an instance. Relatedly, this would be the justification theorem if the definition of ⊤ were dftru2 1542. (Contributed by BJ, 7-Sep-2022.)

Assertion

Ref	Expression
monothetic	⊢ ((𝜑 → 𝜑) ↔ (𝜓 → 𝜓))

Proof of Theorem monothetic

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝜑 → 𝜑)
2		id 22	. 2 ⊢ (𝜓 → 𝜓)
3	1, 2	2th 264	1 ⊢ ((𝜑 → 𝜑) ↔ (𝜓 → 𝜓))

Syntax hints:   → wi 4   ↔ 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:  trujust  1539