Theorem ifpbi123d 1005

Description: Equivalence deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 17-Apr-2024.)

Hypotheses

Ref	Expression
ifpbi123d.1	⊢ (𝜑 → (𝜓 ↔ 𝜏))
ifpbi123d.2	⊢ (𝜑 → (𝜒 ↔ 𝜂))
ifpbi123d.3	⊢ (𝜑 → (𝜃 ↔ 𝜁))

Assertion

Ref	Expression
ifpbi123d	⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))

Proof of Theorem ifpbi123d

Step	Hyp	Ref	Expression
1		ifpbi123d.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜏))
2		ifpbi123d.2	. . . 4 ⊢ (𝜑 → (𝜒 ↔ 𝜂))
3	1, 2	anbi12d 477	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜏 ∧ 𝜂)))
4	1	notbid 677	. . . 4 ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜏))
5		ifpbi123d.3	. . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜁))
6	4, 5	anbi12d 477	. . 3 ⊢ (𝜑 → ((¬ 𝜓 ∧ 𝜃) ↔ (¬ 𝜏 ∧ 𝜁)))
7	3, 6	orbi12d 805	. 2 ⊢ (𝜑 → (((𝜓 ∧ 𝜒) ∨ (¬ 𝜓 ∧ 𝜃)) ↔ ((𝜏 ∧ 𝜂) ∨ (¬ 𝜏 ∧ 𝜁))))
8		df-ifp 991	. 2 ⊢ (if-(𝜓, 𝜒, 𝜃) ↔ ((𝜓 ∧ 𝜒) ∨ (¬ 𝜓 ∧ 𝜃)))
9		df-ifp 991	. 2 ⊢ (if-(𝜏, 𝜂, 𝜁) ↔ ((𝜏 ∧ 𝜂) ∨ (¬ 𝜏 ∧ 𝜁)))
10	7, 8, 9	3bitr4g 223	1 ⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  if-wif 990

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 623  ax-in2 624  ax-io 721

This theorem depends on definitions:  df-bi 117  df-ifp 991

This theorem is referenced by:  ifpbi23d  1006  wkslem1  16544  wkslem2  16545  iswlk  16547  wlkres  16603