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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 r19.29an
Description: A commonly used pattern in the spirit of r19.29 . (Contributed by Thierry Arnoux , 29-Dec-2019) (Proof shortened by Wolf Lammen , 17-Jun-2023)
Ref
Expression
Hypothesis
rexlimdva2.1 ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) → 𝜒 )
Assertion
r19.29an ⊢ ( ( 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 𝜓 ) → 𝜒 )
Proof
Step
Hyp
Ref
Expression
1
rexlimdva2.1 ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) → 𝜒 )
2
1
rexlimdva2 ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜒 ) )
3
2
imp ⊢ ( ( 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 𝜓 ) → 𝜒 )