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Theorem wl-ifpimpr 34876
Description: If one case of an if- condition is a consequence of the other, the expression in df-ifp 1059 can be shortened. (Contributed by Wolf Lammen, 12-Jun-2024.)
Assertion
Ref Expression
wl-ifpimpr ((𝜒𝜓) → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ 𝜒)))

Proof of Theorem wl-ifpimpr
StepHypRef Expression
1 pm4.72 947 . . . . . . 7 ((𝜒𝜓) ↔ (𝜓 ↔ (𝜒𝜓)))
21biimpi 219 . . . . . 6 ((𝜒𝜓) → (𝜓 ↔ (𝜒𝜓)))
3 orcom 867 . . . . . 6 ((𝜒𝜓) ↔ (𝜓𝜒))
42, 3syl6bb 290 . . . . 5 ((𝜒𝜓) → (𝜓 ↔ (𝜓𝜒)))
54anbi2d 631 . . . 4 ((𝜒𝜓) → ((𝜑𝜓) ↔ (𝜑 ∧ (𝜓𝜒))))
6 andi 1005 . . . 4 ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∨ (𝜑𝜒)))
75, 6syl6bb 290 . . 3 ((𝜒𝜓) → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜒))))
87orbi1d 914 . 2 ((𝜒𝜓) → (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ (((𝜑𝜓) ∨ (𝜑𝜒)) ∨ (¬ 𝜑𝜒))))
9 df-ifp 1059 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
10 biidd 265 . . . . 5 (𝜑 → (𝜒𝜒))
11 biidd 265 . . . . 5 𝜑 → (𝜒𝜒))
1210, 11cases 1038 . . . 4 (𝜒 ↔ ((𝜑𝜒) ∨ (¬ 𝜑𝜒)))
1312orbi2i 910 . . 3 (((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜓) ∨ ((𝜑𝜒) ∨ (¬ 𝜑𝜒))))
14 orass 919 . . 3 ((((𝜑𝜓) ∨ (𝜑𝜒)) ∨ (¬ 𝜑𝜒)) ↔ ((𝜑𝜓) ∨ ((𝜑𝜒) ∨ (¬ 𝜑𝜒))))
1513, 14bitr4i 281 . 2 (((𝜑𝜓) ∨ 𝜒) ↔ (((𝜑𝜓) ∨ (𝜑𝜒)) ∨ (¬ 𝜑𝜒)))
168, 9, 153bitr4g 317 1 ((𝜒𝜓) → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ 𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  if-wif 1058
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 400  df-or 845  df-ifp 1059
This theorem is referenced by:  wl-ifp4impr  34877  wl-df2-3mintru2  34895
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