Theorem ifpbi23d 999

Description: Equivalence deduction for conditional operator for propositions. Convenience theorem for a frequent case. (Contributed by Wolf Lammen, 28-Apr-2024.)

Hypotheses

Ref	Expression
ifpbi23d.1	⊢ (𝜑 → (𝜒 ↔ 𝜂))
ifpbi23d.2	⊢ (𝜑 → (𝜃 ↔ 𝜁))

Assertion

Ref	Expression
ifpbi23d	⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜓, 𝜂, 𝜁)))

Proof of Theorem ifpbi23d

Step	Hyp	Ref	Expression
1		biidd 172	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜓))
2		ifpbi23d.1	. 2 ⊢ (𝜑 → (𝜒 ↔ 𝜂))
3		ifpbi23d.2	. 2 ⊢ (𝜑 → (𝜃 ↔ 𝜁))
4	1, 2, 3	ifpbi123d 998	1 ⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜓, 𝜂, 𝜁)))

Syntax hints:   → wi 4   ↔ wb 105  if-wif 983

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 617  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117  df-ifp 984

This theorem is referenced by: (None)