Theorem ifbieq12d2 4542

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 4541	. 2 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
4		ifbieq12d2.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
5	4	ifbid 4531	. 2 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐷) = if(𝜒, 𝐶, 𝐷))
6	3, 5	eqtrd 2769	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  ifcif 4507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3421  df-v 3466  df-un 3938  df-if 4508

This theorem is referenced by:  ofccat  14991  itgeq12dv  34269  sgnneg  34484