Theorem ififcom 33218

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 466	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
2		ifbi 4508	. . 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 4539	. 2 ⊢ if((𝜑 ∧ 𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵)
5		ifan 4539	. 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)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ifcif 4485

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-if 4486

This theorem is used by:  mplasclco  34218