Theorem ifpid 1070

Description: Value of the conditional operator for propositions when the same proposition is returned in either case. Analogue for propositions of ifid 4506. This is essentially pm4.42 1048. (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 1068	. 2 ⊢ (𝜑 → (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓))
2		ifpfal 1069	. 2 ⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓))
3	1, 2	pm2.61i 184	1 ⊢ (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓)

Syntax hints:   ↔ wb 208  if-wif 1057

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 209  df-an 399  df-or 844  df-ifp 1058

This theorem is referenced by: (None)