Theorem ifeq123d 39207

Description: Equality deduction for conditional operator. (Contributed by Glauco Siliprandi, 11-Dec-2019.) AV: This theorem already exists as ifbieq12d 4113. TODO (NM): Please replace the usage of this theorem by ifbieq12d 4113 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 4113	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐷))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483  ifcif 4086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-un 3579  df-if 4087

This theorem is referenced by:  icccncfext  40100  fourierdlem103  40426  fourierdlem104  40427