Theorem ifpnot23d 37830

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

Assertion

Ref	Expression
ifpnot23d	⊢ (¬ if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(𝜑, 𝜓, 𝜒))

Proof of Theorem ifpnot23d

Step	Hyp	Ref	Expression
1		ifpnot23 37823	. 2 ⊢ (¬ if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ ¬ 𝜒))
2		notnotb 304	. . 3 ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
3		notnotb 304	. . 3 ⊢ (𝜒 ↔ ¬ ¬ 𝜒)
4		ifpbi23 37817	. . 3 ⊢ (((𝜓 ↔ ¬ ¬ 𝜓) ∧ (𝜒 ↔ ¬ ¬ 𝜒)) → (if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ ¬ 𝜒)))
5	2, 3, 4	mp2an 708	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ ¬ 𝜒))
6	1, 5	bitr4i 267	1 ⊢ (¬ if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(𝜑, 𝜓, 𝜒))

Syntax hints:  ¬ wn 3   ↔ wb 196  if-wif 1012

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 197  df-or 385  df-an 386  df-ifp 1013

This theorem is referenced by:  ifpororb  37850