Theorem ififcom

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

	Ref	Expression
Assertion	ififcom	$$⊢if\left(\phi ,if\left(\psi ,A,B\right),B\right)=if\left(\psi ,if\left(\phi ,A,B\right),B\right)$$

Proof

Step	Hyp	Ref	Expression
1		ancom	$$⊢\left(\phi \wedge \psi \right)\leftrightarrow \left(\psi \wedge \phi \right)$$
2		ifbi	$$⊢\left(\left(\phi \wedge \psi \right)\leftrightarrow \left(\psi \wedge \phi \right)\right)\to if\left(\left(\phi \wedge \psi \right),A,B\right)=if\left(\left(\psi \wedge \phi \right),A,B\right)$$
3	1 2	ax-mp	$$⊢if\left(\left(\phi \wedge \psi \right),A,B\right)=if\left(\left(\psi \wedge \phi \right),A,B\right)$$
4		ifan	$$⊢if\left(\left(\phi \wedge \psi \right),A,B\right)=if\left(\phi ,if\left(\psi ,A,B\right),B\right)$$
5		ifan	$$⊢if\left(\left(\psi \wedge \phi \right),A,B\right)=if\left(\psi ,if\left(\phi ,A,B\right),B\right)$$
6	3 4 5	3eqtr3i	$$⊢if\left(\phi ,if\left(\psi ,A,B\right),B\right)=if\left(\psi ,if\left(\phi ,A,B\right),B\right)$$