Theorem ifpnot23b 44268

Description: Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)

Assertion

Ref	Expression
ifpnot23b	⊢ (¬ if-(𝜑, ¬ 𝜓, 𝜒) ↔ if-(𝜑, 𝜓, ¬ 𝜒))

Proof of Theorem ifpnot23b

Step	Hyp	Ref	Expression
1		ifpnot23 44264	. 2 ⊢ (¬ if-(𝜑, ¬ 𝜓, 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ 𝜒))
2		notnotb 318	. . 3 ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
3		ifpbi2 44253	. . 3 ⊢ ((𝜓 ↔ ¬ ¬ 𝜓) → (if-(𝜑, 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ 𝜒)))
4	2, 3	ax-mp 5	. 2 ⊢ (if-(𝜑, 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ 𝜒))
5	1, 4	bitr4i 281	1 ⊢ (¬ if-(𝜑, ¬ 𝜓, 𝜒) ↔ if-(𝜑, 𝜓, ¬ 𝜒))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  if-wif 1078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079

This theorem is used by:  ifpbiidcor2  44269