Theorem ifpbi123d 1079

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	imbi12d 348	. . 3 ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜏 → 𝜂)))
4		ifpbi123d.3	. . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜁))
5	1, 4	orbi12d 918	. . 3 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜏 ∨ 𝜁)))
6	3, 5	anbi12d 634	. 2 ⊢ (𝜑 → (((𝜓 → 𝜒) ∧ (𝜓 ∨ 𝜃)) ↔ ((𝜏 → 𝜂) ∧ (𝜏 ∨ 𝜁))))
7		dfifp3 1065	. 2 ⊢ (if-(𝜓, 𝜒, 𝜃) ↔ ((𝜓 → 𝜒) ∧ (𝜓 ∨ 𝜃)))
8		dfifp3 1065	. 2 ⊢ (if-(𝜏, 𝜂, 𝜁) ↔ ((𝜏 → 𝜂) ∧ (𝜏 ∨ 𝜁)))
9	6, 7, 8	3bitr4g 317	1 ⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 846  if-wif 1062

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 400  df-or 847  df-ifp 1063

This theorem is referenced by:  ifpbi23d  1081  wkslem1  27561  wkslem2  27562  iswlk  27564  wlkres  27624  redwlk  27626  wlkp1lem8  27634  crctcshwlkn0lem4  27763  crctcshwlkn0lem5  27764  crctcshwlkn0lem6  27765  1wlkdlem4  28089  pfxwlk  32668  satfv1fvfmla1  32968  ifpbi123  40691