Theorem ifptru 1043

Description: Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4125. This is essentially dedlema 1014. (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 349	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))
2		orc 399	. . . 4 ⊢ (𝜑 → (𝜑 ∨ 𝜒))
3	2	biantrud 527	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒))))
4		dfifp3 1035	. . 3 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))
5	3, 4	syl6bbr 278	. 2 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ if-(𝜑, 𝜓, 𝜒)))
6	1, 5	bitr2d 269	1 ⊢ (𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383  if-wif 1032

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 197  df-or 384  df-an 385  df-ifp 1033

This theorem is referenced by:  ifpfal  1044  ifpid  1045  elimh  1050  dedt  1051  wlkl1loop  26590  lfgrwlkprop  26640  eupth2lem3lem3  27208