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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	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∃ 𝑦 ∃ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		alcom	⊢ ( ∀ 𝑥 ∀ 𝑦 ¬ 𝜑 ↔ ∀ 𝑦 ∀ 𝑥 ¬ 𝜑 )
2	1	notbii	⊢ ( ¬ ∀ 𝑥 ∀ 𝑦 ¬ 𝜑 ↔ ¬ ∀ 𝑦 ∀ 𝑥 ¬ 𝜑 )
3		2exnaln	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ¬ ∀ 𝑥 ∀ 𝑦 ¬ 𝜑 )
4		2exnaln	⊢ ( ∃ 𝑦 ∃ 𝑥 𝜑 ↔ ¬ ∀ 𝑦 ∀ 𝑥 ¬ 𝜑 )
5	2 3 4	3bitr4i	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∃ 𝑦 ∃ 𝑥 𝜑 )