Theorem ifptru 1086

Description: Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4486. This is essentially dedlema 1057. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)

Assertion

Ref	Expression
ifptru	⊢ (𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓))

Proof of Theorem ifptru

Step	Hyp	Ref	Expression
1		biimt 362	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))
2		orc 878	. . . 4 ⊢ (𝜑 → (𝜑 ∨ 𝜒))
3	2	biantrud 539	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒))))
4		dfifp3 1077	. . 3 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))
5	3, 4	bitr4di 291	. 2 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ if-(𝜑, 𝜓, 𝜒)))
6	1, 5	bitr2d 282	1 ⊢ (𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858  if-wif 1074

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 400  df-or 859  df-ifp 1075

This theorem is referenced by:  ifpfal  1087  ifpid  1088  elimh  1094  dedt  1095  axprlem3  5382  axpr  5384  axprlem3OLD  5386  axprlem4OLD  5387  wlkl1loop  29835  lfgrwlkprop  29883  eupth2lem3lem3  30429  axprALT2  35402  satfv1lem  35709  wl-3xortru  37962  sn-axprlem3  42834