Theorem ififcom 32862

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 465	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
2		ifbi 4509	. . 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 4540	. 2 ⊢ if((𝜑 ∧ 𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵)
5		ifan 4540	. 2 ⊢ if((𝜓 ∧ 𝜑), 𝐴, 𝐵) = if(𝜓, if(𝜑, 𝐴, 𝐵), 𝐵)
6	3, 4, 5	3eqtr3i 2792	1 ⊢ if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵) = if(𝜓, if(𝜑, 𝐴, 𝐵), 𝐵)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1568  ifcif 4486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-if 4487

This theorem is referenced by:  mplasclco  33872