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Metamath Proof Explorer

Theorem 19.29r2

Description: Variation of 19.29r with double quantification. (Contributed by NM, 3-Feb-2005)

Ref Expression
Assertion 19.29r2 ( ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∀ 𝑥 ∀ 𝑦 𝜓 ) → ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 19.29r ⊢ ( ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∀ 𝑥 ∀ 𝑦 𝜓 ) → ∃ 𝑥 ( ∃ 𝑦 𝜑 ∧ ∀ 𝑦 𝜓 ) )
2 19.29r ⊢ ( ( ∃ 𝑦 𝜑 ∧ ∀ 𝑦 𝜓 ) → ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) )
3 2 eximi ⊢ ( ∃ 𝑥 ( ∃ 𝑦 𝜑 ∧ ∀ 𝑦 𝜓 ) → ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) )
4 1 3 syl ⊢ ( ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∀ 𝑥 ∀ 𝑦 𝜓 ) → ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) )