Theorem ifeq123d 38231

Description: Equality deduction for conditional operator. (Contributed by Glauco Siliprandi, 11-Dec-2019.) AV: This theorem already exists as ifbieq12d 4063. TODO (NM): Please replace the usage of this theorem by ifbieq12d 4063 then delete this theorem. (New usage is discouraged.)

Hypotheses

Ref	Expression
ifeq123d.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
ifeq123d.2	⊢ (𝜑 → 𝐴 = 𝐵)
ifeq123d.3	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
ifeq123d	⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐷))

Proof of Theorem ifeq123d

Step	Hyp	Ref	Expression
1		ifeq123d.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		ifeq123d.2	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		ifeq123d.3	. 2 ⊢ (𝜑 → 𝐶 = 𝐷)
4	1, 2, 3	ifbieq12d 4063	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐷))

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ifcif 4036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-un 3545  df-if 4037

This theorem is referenced by:  icccncfext  38773  fourierdlem103  39102  fourierdlem104  39103