Theorem ifpnOLD 1071

Description: Obsolete version of ifpn 1070 as of 5-May-2024. (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem ifpnOLD

Step	Hyp	Ref	Expression
1		notnotb 319	. . . 4 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
2	1	imbi1i 354	. . 3 ⊢ ((𝜑 → 𝜓) ↔ (¬ ¬ 𝜑 → 𝜓))
3	2	anbi2ci 628	. 2 ⊢ (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)) ↔ ((¬ 𝜑 → 𝜒) ∧ (¬ ¬ 𝜑 → 𝜓)))
4		dfifp2 1061	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
5		dfifp2 1061	. 2 ⊢ (if-(¬ 𝜑, 𝜒, 𝜓) ↔ ((¬ 𝜑 → 𝜒) ∧ (¬ ¬ 𝜑 → 𝜓)))
6	3, 4, 5	3bitr4i 307	1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  if-wif 1059

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 401  df-or 846  df-ifp 1060

This theorem is referenced by: (None)