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	$$⊢\exists x\exists y\phi \leftrightarrow \exists y\exists x\phi$$

Proof

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