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Theorem ra5 2874
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 1492. (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 2328 . . . 4 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
2 bi2.04 241 . . . . 5 ((𝑥𝐴 → (𝜑𝜓)) ↔ (𝜑 → (𝑥𝐴𝜓)))
32albii 1375 . . . 4 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) ↔ ∀𝑥(𝜑 → (𝑥𝐴𝜓)))
41, 3bitri 177 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝜑 → (𝑥𝐴𝜓)))
5 ra5.1 . . . 4 𝑥𝜑
65stdpc5 1492 . . 3 (∀𝑥(𝜑 → (𝑥𝐴𝜓)) → (𝜑 → ∀𝑥(𝑥𝐴𝜓)))
74, 6sylbi 118 . 2 (∀𝑥𝐴 (𝜑𝜓) → (𝜑 → ∀𝑥(𝑥𝐴𝜓)))
8 df-ral 2328 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
97, 8syl6ibr 155 1 (∀𝑥𝐴 (𝜑𝜓) → (𝜑 → ∀𝑥𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257  wnf 1365  wcel 1409  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-ral 2328
This theorem is referenced by: (None)
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