Theorem alcomiw

Description: Weak version of ax-11 . See alcomw for the biconditional form. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017) (Proof shortened by Wolf Lammen, 28-Dec-2023)

	Ref	Expression
Hypothesis	alcomiw.1	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	alcomiw	$$⊢\forall x\forall y\phi \to \forall y\forall x\phi$$

Proof

Step	Hyp	Ref	Expression
1		alcomiw.1	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
2	1	cbvalvw	$$⊢\forall y\phi \leftrightarrow \forall z\psi$$
3	2	biimpi	$$⊢\forall y\phi \to \forall z\psi$$
4	3	alimi	$$⊢\forall x\forall y\phi \to \forall x\forall z\psi$$
5		ax-5	$$⊢\forall x\forall z\psi \to \forall y\forall x\forall z\psi$$
6	1	biimprd	$$⊢y=z\to \left(\psi \to \phi \right)$$
7	6	equcoms	$$⊢z=y\to \left(\psi \to \phi \right)$$
8	7	spimvw	$$⊢\forall z\psi \to \phi$$
9	8	2alimi	$$⊢\forall y\forall x\forall z\psi \to \forall y\forall x\phi$$
10	4 5 9	3syl	$$⊢\forall x\forall y\phi \to \forall y\forall x\phi$$