Theorem excomimw

Description: Weak version of excomim . Uses only Tarski's FOL axiom schemes. (Contributed by BTernaryTau, 23-Jun-2025)

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

Proof

Step	Hyp	Ref	Expression
1		excomimw.1	$$⊢x=z\to \left(\phi \leftrightarrow \psi \right)$$
2	1	notbid	$$⊢x=z\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
3	2	alcomimw	$$⊢\forall y\forall x\neg \phi \to \forall x\forall y\neg \phi$$
4	3	con3i	$$⊢\neg \forall x\forall y\neg \phi \to \neg \forall y\forall x\neg \phi$$
5		2exnaln	$$⊢\exists x\exists y\phi \leftrightarrow \neg \forall x\forall y\neg \phi$$
6		2exnaln	$$⊢\exists y\exists x\phi \leftrightarrow \neg \forall y\forall x\neg \phi$$
7	4 5 6	3imtr4i	$$⊢\exists x\exists y\phi \to \exists y\exists x\phi$$