Theorem ifpbi123d 1077

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 344	. . 3 ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜏 → 𝜂)))
4		ifpbi123d.3	. . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜁))
5	1, 4	orbi12d 916	. . 3 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜏 ∨ 𝜁)))
6	3, 5	anbi12d 630	. 2 ⊢ (𝜑 → (((𝜓 → 𝜒) ∧ (𝜓 ∨ 𝜃)) ↔ ((𝜏 → 𝜂) ∧ (𝜏 ∨ 𝜁))))
7		dfifp3 1063	. 2 ⊢ (if-(𝜓, 𝜒, 𝜃) ↔ ((𝜓 → 𝜒) ∧ (𝜓 ∨ 𝜃)))
8		dfifp3 1063	. 2 ⊢ (if-(𝜏, 𝜂, 𝜁) ↔ ((𝜏 → 𝜂) ∧ (𝜏 ∨ 𝜁)))
9	6, 7, 8	3bitr4g 314	1 ⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844  if-wif 1060

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 206  df-an 396  df-or 845  df-ifp 1061

This theorem is referenced by:  ifpbi23d  1079  wkslem1  29296  wkslem2  29297  iswlk  29299  wlkres  29359  redwlk  29361  wlkp1lem8  29369  crctcshwlkn0lem4  29499  crctcshwlkn0lem5  29500  crctcshwlkn0lem6  29501  1wlkdlem4  29825  pfxwlk  34577  satfv1fvfmla1  34877  ifpbi123  42703