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Metamath Proof Explorer

Theorem 2r19.29

Description: Theorem r19.29 with two quantifiers. (Contributed by Rodolfo Medina, 25-Sep-2010)

Ref Expression
Assertion 2r19.29 ( ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 r19.29 ⊢ ( ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∃ 𝑦 ∈ 𝐵 𝜓 ) )
2 r19.29 ⊢ ( ( ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∃ 𝑦 ∈ 𝐵 𝜓 ) → ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) )
3 2 reximi ⊢ ( ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∃ 𝑦 ∈ 𝐵 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) )
4 1 3 syl ⊢ ( ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) )