Theorem ifbieq12d2 4517

Description: Equivalence deduction for conditional operators. (Contributed by Thierry Arnoux, 14-Feb-2017.) (Proof shortened by Wolf Lammen, 24-Jun-2021.)

Hypotheses

Ref	Expression
ifbieq12d2.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
ifbieq12d2.2	⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)
ifbieq12d2.3	⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)

Assertion

Ref	Expression
ifbieq12d2	⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))

Proof of Theorem ifbieq12d2

Step	Hyp	Ref	Expression
1		ifbieq12d2.2	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)
2		ifbieq12d2.3	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)
3	1, 2	ifeq12da 4516	. 2 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
4		ifbieq12d2.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
5	4	ifbid 4506	. 2 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐷) = if(𝜒, 𝐶, 𝐷))
6	3, 5	eqtrd 2795	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ifcif 4482

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-un 3904  df-if 4483

This theorem is used by:  ofccat  15042  sgnneg  15173  itgeq12dv  34837