Theorem ifptru 1091

Description: Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4493. This is essentially dedlema 1061. (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 363	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))
2		orc 880	. . . 4 ⊢ (𝜑 → (𝜑 ∨ 𝜒))
3	2	biantrud 540	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒))))
4		dfifp3 1081	. . 3 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))
5	3, 4	bitr4di 292	. 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 860  if-wif 1078

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 861  df-ifp 1079

This theorem is referenced by:  ifpfal  1092  ifpid  1093  elimh  1099  dedt  1100  axprlem3  5396  axpr  5398  axprlem3OLD  5400  axprlem4OLD  5401  wlkl1loop  29987  lfgrwlkprop  30035  eupth2lem3lem3  30581  axprALT2  35503  satfv1lem  35854  wl-3xortru  38117  sn-axprlem3  42989