Theorem alcomw

Description: Weak version of alcom and biconditional form of alcomimw . Uses only Tarski's FOL axiom schemes. (Contributed by BTernaryTau, 28-Dec-2024)

	Ref	Expression
Hypotheses	alcomw.1	$$⊢x=w\to \left(\phi \leftrightarrow \psi \right)$$
	alcomw.2	$$⊢y=z\to \left(\phi \leftrightarrow \chi \right)$$
Assertion	alcomw	$$⊢\forall x\forall y\phi \leftrightarrow \forall y\forall x\phi$$

Proof

Step	Hyp	Ref	Expression
1		alcomw.1	$$⊢x=w\to \left(\phi \leftrightarrow \psi \right)$$
2		alcomw.2	$$⊢y=z\to \left(\phi \leftrightarrow \chi \right)$$
3	2	alcomimw	$$⊢\forall x\forall y\phi \to \forall y\forall x\phi$$
4	1	alcomimw	$$⊢\forall y\forall x\phi \to \forall x\forall y\phi$$
5	3 4	impbii	$$⊢\forall x\forall y\phi \leftrightarrow \forall y\forall x\phi$$