Theorem ifpnst 997

Description: Conditional operator for the negation of a proposition. (Contributed by BJ, 30-Sep-2019.) (Proof shortened by Wolf Lammen, 5-May-2024.)

Assertion

Ref	Expression
ifpnst	⊢ (STAB 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓)))

Proof of Theorem ifpnst

Step	Hyp	Ref	Expression
1		ifpdc 988	. . 3 ⊢ (if-(𝜑, 𝜓, 𝜒) → DECID 𝜑)
2	1	adantl 277	. 2 ⊢ ((STAB 𝜑 ∧ if-(𝜑, 𝜓, 𝜒)) → DECID 𝜑)
3		ifpdc 988	. . 3 ⊢ (if-(¬ 𝜑, 𝜒, 𝜓) → DECID ¬ 𝜑)
4		stdcndc 853	. . . 4 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) ↔ DECID 𝜑)
5	4	biimpi 120	. . 3 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → DECID 𝜑)
6	3, 5	sylan2 286	. 2 ⊢ ((STAB 𝜑 ∧ if-(¬ 𝜑, 𝜒, 𝜓)) → DECID 𝜑)
7		dfifp5dc 993	. . . 4 ⊢ (DECID 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (¬ 𝜑 → 𝜒))))
8	7	biancomd 271	. . 3 ⊢ (DECID 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 → 𝜒) ∧ (¬ 𝜑 ∨ 𝜓))))
9		dcn 850	. . . 4 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
10		dfifp3dc 991	. . . 4 ⊢ (DECID ¬ 𝜑 → (if-(¬ 𝜑, 𝜒, 𝜓) ↔ ((¬ 𝜑 → 𝜒) ∧ (¬ 𝜑 ∨ 𝜓))))
11	9, 10	syl 14	. . 3 ⊢ (DECID 𝜑 → (if-(¬ 𝜑, 𝜒, 𝜓) ↔ ((¬ 𝜑 → 𝜒) ∧ (¬ 𝜑 ∨ 𝜓))))
12	8, 11	bitr4d 191	. 2 ⊢ (DECID 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓)))
13	2, 6, 12	pm5.21nd 924	1 ⊢ (STAB 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716  STAB wstab 838  DECID wdc 842  if-wif 986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-ifp 987

This theorem is referenced by: (None)