Theorem wl-ifp4impr 37844

Description: If one case of an if- condition is a consequence of the other, the expression in dfifp4 1073 can be shortened. (Contributed by Wolf Lammen, 18-Jun-2024.)

Assertion

Ref	Expression
wl-ifp4impr	⊢ ((𝜒 → 𝜓) → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ 𝜓)))

Proof of Theorem wl-ifp4impr

Step	Hyp	Ref	Expression
1		wl-ifpimpr 37843	. 2 ⊢ ((𝜒 → 𝜓) → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ 𝜒)))
2		pm4.71 563	. . . . 5 ⊢ ((𝜒 → 𝜓) ↔ (𝜒 ↔ (𝜒 ∧ 𝜓)))
3	2	biimpi 218	. . . 4 ⊢ ((𝜒 → 𝜓) → (𝜒 ↔ (𝜒 ∧ 𝜓)))
4	3	orbi2d 922	. . 3 ⊢ ((𝜒 → 𝜓) → (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜓))))
5		andir 1017	. . 3 ⊢ (((𝜑 ∨ 𝜒) ∧ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜓)))
6	4, 5	bitr4di 291	. 2 ⊢ ((𝜒 → 𝜓) → (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ 𝜓)))
7	1, 6	bitrd 281	1 ⊢ ((𝜒 → 𝜓) → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∨ wo 854  if-wif 1069

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 209  df-an 398  df-or 855  df-ifp 1070

This theorem is referenced by:  wl-df3-3mintru2  37863