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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 r19.37
Description: Restricted quantifier version of one direction of 19.37 . (The other
direction does not hold when A is empty.) (Contributed by FL , 13-May-2012) (Revised by Mario Carneiro , 11-Dec-2016)
Ref
Expression
Hypothesis
r19.37.1 ⊢ Ⅎ 𝑥 𝜑
Assertion
r19.37 ⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
Proof
Step
Hyp
Ref
Expression
1
r19.37.1 ⊢ Ⅎ 𝑥 𝜑
2
r19.35 ⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
3
ax-1 ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜑 ) )
4
1 3
ralrimi ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜑 )
5
4
imim1i ⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) → ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
6
2 5
sylbi ⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )