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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-4 (Quantified Implication) alrimdh
Description: Deduction form of Theorem 19.21 of Margaris p. 90, see 19.21 and
19.21h . (Contributed by NM , 10-Feb-1997) (Proof shortened by Andrew Salmon , 13-May-2011)
Ref
Expression
Hypotheses
alrimdh.1 ⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
alrimdh.2 ⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
alrimdh.3 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
Assertion
alrimdh ⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 𝜒 ) )
Proof
Step
Hyp
Ref
Expression
1
alrimdh.1 ⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2
alrimdh.2 ⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
3
alrimdh.3 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
4
1 3
alimdh ⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑥 𝜒 ) )
5
2 4
syl5 ⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 𝜒 ) )