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Database ZF (ZERMELO-FRAENKEL) SET THEORY ZF Set Theory - start with the Axiom of Extensionality Restricted quantification Restricted universal and existential quantification r19.29
Description: Restricted quantifier version of 19.29 . See also r19.29r .
(Contributed by NM , 31-Aug-1999) (Proof shortened by Andrew Salmon , 30-May-2011) (Proof shortened by Wolf Lammen , 22-Dec-2024)
Ref
Expression
Assertion
r19.29 ⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) )
Proof
Step
Hyp
Ref
Expression
1
ibar ⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜑 ∧ 𝜓 ) ) )
2
1
ralrexbid ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ) )
3
2
biimpa ⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) )