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Database CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes) Axiom scheme ax-5 (Distinctness) - first use of $d 19.36imv
Description: One direction of 19.36v that can be proven without ax-6 .
(Contributed by Rohan Ridenour , 16-Apr-2022) (Proof shortened by Wolf
Lammen , 22-Sep-2024)
Ref
Expression
Assertion
19.36imv ⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → 𝜓 ) )
Proof
Step
Hyp
Ref
Expression
1
pm2.27 ⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) )
2
1
aleximi ⊢ ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ∃ 𝑥 𝜓 ) )
3
ax5e ⊢ ( ∃ 𝑥 𝜓 → 𝜓 )
4
2 3
syl6com ⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → 𝜓 ) )