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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 r19.45v
Description: Restricted quantifier version of one direction of 19.45 . The other
direction holds when A is nonempty, see r19.45zv . (Contributed by NM , 2-Apr-2004)
Ref
Expression
Assertion
r19.45v ⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) )
Proof
Step
Hyp
Ref
Expression
1
r19.43 ⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) )
2
id ⊢ ( 𝜑 → 𝜑 )
3
2
rexlimivw ⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝜑 )
4
3
orim1i ⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) → ( 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) )
5
1 4
sylbi ⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) )