Theorem ifpnot23 44478

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 997	. . . 4 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
2		pm4.55 1003	. . . 4 ⊢ (¬ (¬ 𝜑 ∧ 𝜒) ↔ (𝜑 ∨ ¬ 𝜒))
3	1, 2	anbi12i 640	. . 3 ⊢ ((¬ (𝜑 ∧ 𝜓) ∧ ¬ (¬ 𝜑 ∧ 𝜒)) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∧ (𝜑 ∨ ¬ 𝜒)))
4		ioran 999	. . 3 ⊢ (¬ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ (¬ (𝜑 ∧ 𝜓) ∧ ¬ (¬ 𝜑 ∧ 𝜒)))
5		dfifp4 1082	. . 3 ⊢ (if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∧ (𝜑 ∨ ¬ 𝜒)))
6	3, 4, 5	3bitr4i 306	. 2 ⊢ (¬ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒))
7		df-ifp 1079	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
8	6, 7	xchnxbir 336	1 ⊢ (¬ if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861  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:  ifpnotnotb  44479  ifpnorcor  44480  ifpnancor  44481  ifpnot23b  44482  ifpnot23c  44484  ifpnot23d  44485  ifpdfnan  44486  ifpdfxor  44487  ifpor123g  44508