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Metamath Proof Explorer

Theorem 19.40-2

Description: Theorem *11.42 in WhiteheadRussell p. 163. Theorem 19.40 of Margaris p. 90 with two quantifiers. (Contributed by Andrew Salmon, 24-May-2011)

		Ref	Expression
	Assertion	19.40-2	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) → ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∃ 𝑥 ∃ 𝑦 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.40	⊢ ( ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) → ( ∃ 𝑦 𝜑 ∧ ∃ 𝑦 𝜓 ) )
2	1	eximi	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) → ∃ 𝑥 ( ∃ 𝑦 𝜑 ∧ ∃ 𝑦 𝜓 ) )
3		19.40	⊢ ( ∃ 𝑥 ( ∃ 𝑦 𝜑 ∧ ∃ 𝑦 𝜓 ) → ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∃ 𝑥 ∃ 𝑦 𝜓 ) )
4	2 3	syl	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) → ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∃ 𝑥 ∃ 𝑦 𝜓 ) )