Theorem ifbieq12d2 4502

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 4501	. 2 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
4		ifbieq12d2.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
5	4	ifbid 4491	. 2 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐷) = if(𝜒, 𝐶, 𝐷))
6	3, 5	eqtrd 2858	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ifcif 4469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-un 3943  df-if 4470

This theorem is referenced by:  ofccat  14331  itgeq12dv  31586  sgnneg  31800