Theorem ifpnot23 40186

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 979	. . . 4 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
2		pm4.55 985	. . . 4 ⊢ (¬ (¬ 𝜑 ∧ 𝜒) ↔ (𝜑 ∨ ¬ 𝜒))
3	1, 2	anbi12i 629	. . 3 ⊢ ((¬ (𝜑 ∧ 𝜓) ∧ ¬ (¬ 𝜑 ∧ 𝜒)) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∧ (𝜑 ∨ ¬ 𝜒)))
4		ioran 981	. . 3 ⊢ (¬ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ (¬ (𝜑 ∧ 𝜓) ∧ ¬ (¬ 𝜑 ∧ 𝜒)))
5		dfifp4 1062	. . 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 1059	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
8	6, 7	xchnxbir 336	1 ⊢ (¬ if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∨ wo 844  if-wif 1058

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 210  df-an 400  df-or 845  df-ifp 1059

This theorem is referenced by:  ifpnotnotb  40187  ifpnorcor  40188  ifpnancor  40189  ifpnot23b  40190  ifpnot23c  40192  ifpnot23d  40193  ifpdfnan  40194  ifpdfxor  40195  ifpor123g  40216