Theorem ifpbi123 36854

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

Assertion

Ref	Expression
ifpbi123	⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜂)))

Proof of Theorem ifpbi123

Step	Hyp	Ref	Expression
1		simp1 1054	. . . 4 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (𝜑 ↔ 𝜓))
2		simp2 1055	. . . 4 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (𝜒 ↔ 𝜃))
3	1, 2	imbi12d 333	. . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))
4	1	notbid 307	. . . 4 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (¬ 𝜑 ↔ ¬ 𝜓))
5		simp3 1056	. . . 4 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (𝜏 ↔ 𝜂))
6	4, 5	imbi12d 333	. . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → ((¬ 𝜑 → 𝜏) ↔ (¬ 𝜓 → 𝜂)))
7	3, 6	anbi12d 743	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (((𝜑 → 𝜒) ∧ (¬ 𝜑 → 𝜏)) ↔ ((𝜓 → 𝜃) ∧ (¬ 𝜓 → 𝜂))))
8		dfifp2 1008	. 2 ⊢ (if-(𝜑, 𝜒, 𝜏) ↔ ((𝜑 → 𝜒) ∧ (¬ 𝜑 → 𝜏)))
9		dfifp2 1008	. 2 ⊢ (if-(𝜓, 𝜃, 𝜂) ↔ ((𝜓 → 𝜃) ∧ (¬ 𝜓 → 𝜂)))
10	7, 8, 9	3bitr4g 302	1 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜂)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  if-wif 1006   ∧ w3a 1031

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 196  df-or 384  df-an 385  df-ifp 1007  df-3an 1033

This theorem is referenced by: (None)