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Theorem ra5 3130
 Description: Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is an axiom of a predicate calculus for a restricted domain. Compare the unrestricted stdpc5 1798. (Contributed by NM, 16-Jan-2004.)
Hypothesis
Ref Expression
ra5.1 xφ
Assertion
Ref Expression
ra5 (x A (φψ) → (φx A ψ))

Proof of Theorem ra5
StepHypRef Expression
1 df-ral 2619 . . . 4 (x A (φψ) ↔ x(x A → (φψ)))
2 bi2.04 350 . . . . 5 ((x A → (φψ)) ↔ (φ → (x Aψ)))
32albii 1566 . . . 4 (x(x A → (φψ)) ↔ x(φ → (x Aψ)))
41, 3bitri 240 . . 3 (x A (φψ) ↔ x(φ → (x Aψ)))
5 ra5.1 . . . 4 xφ
65stdpc5 1798 . . 3 (x(φ → (x Aψ)) → (φx(x Aψ)))
74, 6sylbi 187 . 2 (x A (φψ) → (φx(x Aψ)))
8 df-ral 2619 . 2 (x A ψx(x Aψ))
97, 8syl6ibr 218 1 (x A (φψ) → (φx A ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540  Ⅎwnf 1544   ∈ wcel 1710  ∀wral 2614 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545  df-ral 2619 This theorem is referenced by: (None)
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