Theorem wl-ifpimpr 35616

Description: If one case of an if- condition is a consequence of the other, the expression in df-ifp 1060 can be shortened. (Contributed by Wolf Lammen, 12-Jun-2024.)

Assertion

Ref	Expression
wl-ifpimpr	⊢ ((𝜒 → 𝜓) → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ 𝜒)))

Proof of Theorem wl-ifpimpr

Step	Hyp	Ref	Expression
1		pm4.72 946	. . . . . . 7 ⊢ ((𝜒 → 𝜓) ↔ (𝜓 ↔ (𝜒 ∨ 𝜓)))
2	1	biimpi 215	. . . . . 6 ⊢ ((𝜒 → 𝜓) → (𝜓 ↔ (𝜒 ∨ 𝜓)))
3		orcom 866	. . . . . 6 ⊢ ((𝜒 ∨ 𝜓) ↔ (𝜓 ∨ 𝜒))
4	2, 3	bitrdi 286	. . . . 5 ⊢ ((𝜒 → 𝜓) → (𝜓 ↔ (𝜓 ∨ 𝜒)))
5	4	anbi2d 628	. . . 4 ⊢ ((𝜒 → 𝜓) → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ (𝜓 ∨ 𝜒))))
6		andi 1004	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)))
7	5, 6	bitrdi 286	. . 3 ⊢ ((𝜒 → 𝜓) → ((𝜑 ∧ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒))))
8	7	orbi1d 913	. 2 ⊢ ((𝜒 → 𝜓) → (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∨ (¬ 𝜑 ∧ 𝜒))))
9		df-ifp 1060	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
10		biidd 261	. . . . 5 ⊢ (𝜑 → (𝜒 ↔ 𝜒))
11		biidd 261	. . . . 5 ⊢ (¬ 𝜑 → (𝜒 ↔ 𝜒))
12	10, 11	cases 1039	. . . 4 ⊢ (𝜒 ↔ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜒)))
13	12	orbi2i 909	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜒))))
14		orass 918	. . 3 ⊢ ((((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜒))))
15	13, 14	bitr4i 277	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∨ (¬ 𝜑 ∧ 𝜒)))
16	8, 9, 15	3bitr4g 313	1 ⊢ ((𝜒 → 𝜓) → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843  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 206  df-an 396  df-or 844  df-ifp 1060

This theorem is referenced by:  wl-ifp4impr  35617  wl-df2-3mintru2  35635