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Metamath Proof Explorer

Theorem excom

Description: Theorem 19.11 of Margaris p. 89. (Contributed by NM, 5-Aug-1993) Remove dependencies on ax-5 , ax-6 , ax-7 , ax-10 , ax-12 . (Revised by Wolf Lammen, 8-Jan-2018) (Proof shortened by Wolf Lammen, 22-Aug-2020)

		Ref	Expression
	Assertion	excom	$⊢ \exists x \exists y φ \leftrightarrow \exists y \exists x φ$

Proof

Step	Hyp	Ref	Expression
1		alcom	$⊢ \forall x \forall y \neg φ \leftrightarrow \forall y \forall x \neg φ$
2	1	notbii	$⊢ \neg \forall x \forall y \neg φ \leftrightarrow \neg \forall y \forall x \neg φ$
3		2exnaln	$⊢ \exists x \exists y φ \leftrightarrow \neg \forall x \forall y \neg φ$
4		2exnaln	$⊢ \exists y \exists x φ \leftrightarrow \neg \forall y \forall x \neg φ$
5	2 3 4	3bitr4i	$⊢ \exists x \exists y φ \leftrightarrow \exists y \exists x φ$