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Theorem ra5 3025
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 1564. (Contributed by NM, 16-Jan-2004.)
Hypothesis
Ref Expression
ra5.1 𝑥𝜑
Assertion
Ref Expression
ra5 (∀𝑥𝐴 (𝜑𝜓) → (𝜑 → ∀𝑥𝐴 𝜓))

Proof of Theorem ra5
StepHypRef Expression
1 df-ral 2440 . . . 4 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
2 bi2.04 247 . . . . 5 ((𝑥𝐴 → (𝜑𝜓)) ↔ (𝜑 → (𝑥𝐴𝜓)))
32albii 1450 . . . 4 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) ↔ ∀𝑥(𝜑 → (𝑥𝐴𝜓)))
41, 3bitri 183 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝜑 → (𝑥𝐴𝜓)))
5 ra5.1 . . . 4 𝑥𝜑
65stdpc5 1564 . . 3 (∀𝑥(𝜑 → (𝑥𝐴𝜓)) → (𝜑 → ∀𝑥(𝑥𝐴𝜓)))
74, 6sylbi 120 . 2 (∀𝑥𝐴 (𝜑𝜓) → (𝜑 → ∀𝑥(𝑥𝐴𝜓)))
8 df-ral 2440 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
97, 8syl6ibr 161 1 (∀𝑥𝐴 (𝜑𝜓) → (𝜑 → ∀𝑥𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1333  wnf 1440  wcel 2128  wral 2435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-4 1490  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-ral 2440
This theorem is referenced by: (None)
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