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Database SUPPLEMENTARY MATERIAL (USERS' MATHBOXES) Mathbox for BJ First-order logic Adding ax-5 bj-cbvalimi
Description: An equality-free general instance of one half of a precise form of
bj-cbval . (Contributed by BJ , 12-Mar-2023)
(Proof modification is discouraged.)
Ref
Expression
Hypotheses
bj-cbvalimi.maj ⊢ ( 𝜒 → ( 𝜑 → 𝜓 ) )
bj-cbvalimi.denote ⊢ ∀ 𝑦 ∃ 𝑥 𝜒
Assertion
bj-cbvalimi ⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 )
Proof
Step
Hyp
Ref
Expression
1
bj-cbvalimi.maj ⊢ ( 𝜒 → ( 𝜑 → 𝜓 ) )
2
bj-cbvalimi.denote ⊢ ∀ 𝑦 ∃ 𝑥 𝜒
3
1
gen2 ⊢ ∀ 𝑦 ∀ 𝑥 ( 𝜒 → ( 𝜑 → 𝜓 ) )
4
bj-cbvalim ⊢ ( ∀ 𝑦 ∃ 𝑥 𝜒 → ( ∀ 𝑦 ∀ 𝑥 ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )
5
2 3 4
mp2 ⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 )