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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 r19.29a
Description: A commonly used pattern in the spirit of r19.29 . (Contributed by Thierry Arnoux , 22-Nov-2017) Reduce axiom usage. (Revised by Wolf
Lammen , 17-Jun-2023)
Ref
Expression
Hypotheses
r19.29a.1 ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) → 𝜒 )
r19.29a.2 ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
Assertion
r19.29a ⊢ ( 𝜑 → 𝜒 )
Proof
Step
Hyp
Ref
Expression
1
r19.29a.1 ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) → 𝜒 )
2
r19.29a.2 ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
3
1
rexlimdva2 ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜒 ) )
4
2 3
mpd ⊢ ( 𝜑 → 𝜒 )