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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 r19.36v
Description: Restricted quantifier version of one direction of 19.36 . (The other
direction holds iff A is nonempty, see r19.36zv .) (Contributed by NM , 22-Oct-2003)
Ref
Expression
Assertion
r19.36v ⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) )
Proof
Step
Hyp
Ref
Expression
1
r19.35 ⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
2
id ⊢ ( 𝜓 → 𝜓 )
3
2
rexlimivw ⊢ ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜓 )
4
3
imim2i ⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) )
5
1 4
sylbi ⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) )