Theorem ifpid 1075

Description: Value of the conditional operator for propositions when the same proposition is returned in either case. Analogue for propositions of ifid 4499. This is essentially pm4.42 1051. (Contributed by BJ, 20-Sep-2019.)

Assertion

Ref	Expression
ifpid	⊢ (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓)

Proof of Theorem ifpid

Step	Hyp	Ref	Expression
1		ifptru 1073	. 2 ⊢ (𝜑 → (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓))
2		ifpfal 1074	. 2 ⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓))
3	1, 2	pm2.61i 182	1 ⊢ (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓)

Syntax hints:   ↔ wb 205  if-wif 1060

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  df-an 397  df-or 845  df-ifp 1061

This theorem is referenced by:  wl-1mintru2  35660