Theorem wl-ifp-ncond2 35563

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) ↔ (𝜑 ∧ 𝜓)))

Proof of Theorem wl-ifp-ncond2

Step	Hyp	Ref	Expression
1		wl-ifp-ncond1 35562	. 2 ⊢ (¬ 𝜒 → (if-(¬ 𝜑, 𝜒, 𝜓) ↔ (¬ ¬ 𝜑 ∧ 𝜓)))
2		ifpn 1070	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
3		notnotb 314	. . 3 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
4	3	anbi1i 623	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ (¬ ¬ 𝜑 ∧ 𝜓))
5	1, 2, 4	3bitr4g 313	1 ⊢ (¬ 𝜒 → (if-(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∧ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  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: (None)