Theorem ifptru 1080

Description: Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4460. This is essentially dedlema 1051. (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 361	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))
2		orc 873	. . . 4 ⊢ (𝜑 → (𝜑 ∨ 𝜒))
3	2	biantrud 536	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒))))
4		dfifp3 1071	. . 3 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))
5	3, 4	bitr4di 290	. 2 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ if-(𝜑, 𝜓, 𝜒)))
6	1, 5	bitr2d 281	1 ⊢ (𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853  if-wif 1068

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 208  df-an 397  df-or 854  df-ifp 1069

This theorem is referenced by:  ifpfal  1081  ifpid  1082  elimh  1088  dedt  1089  axprlem3  5354  axpr  5356  axprlem3OLD  5358  axprlem4OLD  5359  wlkl1loop  29724  lfgrwlkprop  29772  eupth2lem3lem3  30318  axprALT2  35290  satfv1lem  35590  wl-3xortru  37833  sn-axprlem3  42705