Theorem ifpbi23d 1096

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 265	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜓))
2		ifpbi23d.1	. 2 ⊢ (𝜑 → (𝜒 ↔ 𝜂))
3		ifpbi23d.2	. 2 ⊢ (𝜑 → (𝜃 ↔ 𝜁))
4	1, 2, 3	ifpbi123d 1095	1 ⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜓, 𝜂, 𝜁)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  if-wif 1078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079

This theorem is used by:  wksfval  30190  subgrwlk  30269  satfv1fvfmla1  36188  bj-ififc  37452  wl-df-3xor  38392  wl-df3maxtru1  38416  ifpbi23  44474