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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 r19.29r
Description: Restricted quantifier version of 19.29r ; variation of r19.29 .
(Contributed by NM , 31-Aug-1999) (Proof shortened by Wolf Lammen , 29-Jun-2023)
Ref
Expression
Assertion
r19.29r ⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) )
Proof
Step
Hyp
Ref
Expression
1
iba ⊢ ( 𝜓 → ( 𝜑 ↔ ( 𝜑 ∧ 𝜓 ) ) )
2
1
ralrexbid ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 → ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ) )
3
2
biimpac ⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) )