Theorem ififcom 32755

Description: Commute two nested conditionals. (Contributed by Thierry Arnoux, 4-May-2026.)

Assertion

Ref	Expression
ififcom	⊢ if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵) = if(𝜓, if(𝜑, 𝐴, 𝐵), 𝐵)

Proof of Theorem ififcom

Step	Hyp	Ref	Expression
1		ancom 464	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
2		ifbi 4504	. . 3 ⊢ (((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑)) → if((𝜑 ∧ 𝜓), 𝐴, 𝐵) = if((𝜓 ∧ 𝜑), 𝐴, 𝐵))
3	1, 2	ax-mp 5	. 2 ⊢ if((𝜑 ∧ 𝜓), 𝐴, 𝐵) = if((𝜓 ∧ 𝜑), 𝐴, 𝐵)
4		ifan 4535	. 2 ⊢ if((𝜑 ∧ 𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵)
5		ifan 4535	. 2 ⊢ if((𝜓 ∧ 𝜑), 𝐴, 𝐵) = if(𝜓, if(𝜑, 𝐴, 𝐵), 𝐵)
6	3, 4, 5	3eqtr3i 2794	1 ⊢ if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵) = if(𝜓, if(𝜑, 𝐴, 𝐵), 𝐵)

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1561  ifcif 4481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-if 4482

This theorem is referenced by:  mplasclco  33815