Theorem ifpnot23 43491

Description: Negation of conditional logical operator. (Contributed by RP, 18-Apr-2020.)

Assertion

Ref	Expression
ifpnot23	⊢ (¬ if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒))

Proof of Theorem ifpnot23

Step	Hyp	Ref	Expression
1		ianor 984	. . . 4 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
2		pm4.55 990	. . . 4 ⊢ (¬ (¬ 𝜑 ∧ 𝜒) ↔ (𝜑 ∨ ¬ 𝜒))
3	1, 2	anbi12i 628	. . 3 ⊢ ((¬ (𝜑 ∧ 𝜓) ∧ ¬ (¬ 𝜑 ∧ 𝜒)) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∧ (𝜑 ∨ ¬ 𝜒)))
4		ioran 986	. . 3 ⊢ (¬ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ (¬ (𝜑 ∧ 𝜓) ∧ ¬ (¬ 𝜑 ∧ 𝜒)))
5		dfifp4 1067	. . 3 ⊢ (if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∧ (𝜑 ∨ ¬ 𝜒)))
6	3, 4, 5	3bitr4i 303	. 2 ⊢ (¬ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒))
7		df-ifp 1064	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
8	6, 7	xchnxbir 333	1 ⊢ (¬ if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 848  if-wif 1063

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 207  df-an 396  df-or 849  df-ifp 1064

This theorem is referenced by:  ifpnotnotb  43492  ifpnorcor  43493  ifpnancor  43494  ifpnot23b  43495  ifpnot23c  43497  ifpnot23d  43498  ifpdfnan  43499  ifpdfxor  43500  ifpor123g  43521