Theorem frege54cor1a 37178

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 37176	. 2 ⊢ (𝜑 ↔ 𝜑)
2		frege54cor0a 37177	. 2 ⊢ ((𝜑 ↔ 𝜑) ↔ if-(𝜑, 𝜑, ¬ 𝜑))
3	1, 2	mpbi 219	1 ⊢ if-(𝜑, 𝜑, ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 195  if-wif 1006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege28 37144  ax-frege54a 37176

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007

This theorem is referenced by:  frege55a  37182