Theorem frege54cor1a 44618

Description: Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege54cor1a	⊢ if-(𝜑, 𝜑, ¬ 𝜑)

Proof of Theorem frege54cor1a

Step	Hyp	Ref	Expression
1		ax-frege54a 44616	. 2 ⊢ (𝜑 ↔ 𝜑)
2		frege54cor0a 44617	. 2 ⊢ ((𝜑 ↔ 𝜑) ↔ if-(𝜑, 𝜑, ¬ 𝜑))
3	1, 2	mpbi 233	1 ⊢ if-(𝜑, 𝜑, ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  if-wif 1078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege28 44584  ax-frege54a 44616

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1079

This theorem is used by:  frege55a  44622