Theorem ifpid 1077

Description: Value of the conditional operator for propositions when the same proposition is returned in either case. Analogue for propositions of ifid 4566. This is essentially pm4.42 1054. (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 1075	. 2 ⊢ (𝜑 → (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓))
2		ifpfal 1076	. 2 ⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓))
3	1, 2	pm2.61i 182	1 ⊢ (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓)

Syntax hints:   ↔ wb 206  if-wif 1063

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  df-an 396  df-or 849  df-ifp 1064

This theorem is referenced by:  wl-1mintru2  37490