Theorem ifpbi13 44489

Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)

Assertion

Ref	Expression
ifpbi13	⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜑, 𝜏, 𝜒) ↔ if-(𝜓, 𝜏, 𝜃)))

Proof of Theorem ifpbi13

Step	Hyp	Ref	Expression
1		simpl 488	. . . 4 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (𝜑 ↔ 𝜓))
2	1	imbi1d 344	. . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((𝜑 → 𝜏) ↔ (𝜓 → 𝜏)))
3		notbi 322	. . . . 5 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
4		imbi12 349	. . . . 5 ⊢ ((¬ 𝜑 ↔ ¬ 𝜓) → ((𝜒 ↔ 𝜃) → ((¬ 𝜑 → 𝜒) ↔ (¬ 𝜓 → 𝜃))))
5	3, 4	sylbi 220	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((¬ 𝜑 → 𝜒) ↔ (¬ 𝜓 → 𝜃))))
6	5	imp 412	. . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((¬ 𝜑 → 𝜒) ↔ (¬ 𝜓 → 𝜃)))
7	2, 6	anbi12d 644	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (((𝜑 → 𝜏) ∧ (¬ 𝜑 → 𝜒)) ↔ ((𝜓 → 𝜏) ∧ (¬ 𝜓 → 𝜃))))
8		dfifp2 1080	. 2 ⊢ (if-(𝜑, 𝜏, 𝜒) ↔ ((𝜑 → 𝜏) ∧ (¬ 𝜑 → 𝜒)))
9		dfifp2 1080	. 2 ⊢ (if-(𝜓, 𝜏, 𝜃) ↔ ((𝜓 → 𝜏) ∧ (¬ 𝜓 → 𝜃)))
10	7, 8, 9	3bitr4g 317	1 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜑, 𝜏, 𝜒) ↔ if-(𝜓, 𝜏, 𝜃)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  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: (None)