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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 ralanid
Description: Cancellation law for restricted universal quantification. (Contributed by Peter Mazsa , 30-Dec-2018) (Proof shortened by Wolf Lammen , 29-Jun-2023)
Ref
Expression
Assertion
ralanid ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∀ 𝑥 ∈ 𝐴 𝜑 )
Proof
Step
Hyp
Ref
Expression
1
ibar ⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
2
1
bicomd ⊢ ( 𝑥 ∈ 𝐴 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ 𝜑 ) )
3
2
ralbiia ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∀ 𝑥 ∈ 𝐴 𝜑 )