Theorem ifptru 1072

Description: Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4427. This is essentially dedlema 1042. (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 365	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))
2		orc 865	. . . 4 ⊢ (𝜑 → (𝜑 ∨ 𝜒))
3	2	biantrud 536	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒))))
4		dfifp3 1062	. . 3 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))
5	3, 4	bitr4di 293	. 2 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ if-(𝜑, 𝜓, 𝜒)))
6	1, 5	bitr2d 283	1 ⊢ (𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 845  if-wif 1059

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 210  df-an 401  df-or 846  df-ifp 1060

This theorem is referenced by:  ifpfal  1073  ifpid  1074  elimh  1081  dedt  1082  axprlem3  5295  axprlem4  5296  wlkl1loop  27519  lfgrwlkprop  27569  eupth2lem3lem3  28107  satfv1lem  32833  wl-3xortru  35161  sn-axprlem3  39693