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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 ralimdvv
Description: Deduction doubly quantifying both antecedent and consequent.
(Contributed by Scott Fenton , 2-Mar-2025) Shorten and reduce DV
conditions. (Revised by Eric Schmidt , 18-Nov-2025)
Ref
Expression
Hypothesis
ralimdvv.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
Assertion
ralimdvv ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜒 ) )
Proof
Step
Hyp
Ref
Expression
1
ralimdvv.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2
1
ralimdv ⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐵 𝜓 → ∀ 𝑦 ∈ 𝐵 𝜒 ) )
3
2
ralimdv ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜒 ) )