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Theorem wl-ifp-ncond2 34949
 Description: If one case of an if- condition is false, the other automatically follows. (Contributed by Wolf Lammen, 21-Jul-2024.)
Assertion
Ref Expression
wl-ifp-ncond2 𝜒 → (if-(𝜑, 𝜓, 𝜒) ↔ (𝜑𝜓)))

Proof of Theorem wl-ifp-ncond2
StepHypRef Expression
1 wl-ifp-ncond1 34948 . 2 𝜒 → (if-(¬ 𝜑, 𝜒, 𝜓) ↔ (¬ ¬ 𝜑𝜓)))
2 ifpn 1069 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
3 notnotb 318 . . 3 (𝜑 ↔ ¬ ¬ 𝜑)
43anbi1i 626 . 2 ((𝜑𝜓) ↔ (¬ ¬ 𝜑𝜓))
51, 2, 43bitr4g 317 1 𝜒 → (if-(𝜑, 𝜓, 𝜒) ↔ (𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  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: (None)
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