Theorem ifeq123d 39706

Description: Equality deduction for conditional operator. (Contributed by Glauco Siliprandi, 11-Dec-2019.) AV: This theorem already exists as ifbieq12d 4257. TODO (NM): Please replace the usage of this theorem by ifbieq12d 4257 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 4257	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐷))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1632  ifcif 4230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-un 3720  df-if 4231

This theorem is referenced by:  icccncfext  40603  fourierdlem103  40929  fourierdlem104  40930