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Theorem hbralrimi 3183
Description: Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). This theorem contains the common proof steps for ralrimi 3219 and ralrimiv 3184. Its main advantage over these two is its minimal references to axioms. The proof is extracted from NM's previous work. (Contributed by Wolf Lammen, 4-Dec-2019.)
Hypotheses
Ref Expression
hbralrimi.1 (𝜑 → ∀𝑥𝜑)
hbralrimi.2 (𝜑 → (𝑥𝐴𝜓))
Assertion
Ref Expression
hbralrimi (𝜑 → ∀𝑥𝐴 𝜓)

Proof of Theorem hbralrimi
StepHypRef Expression
1 hbralrimi.1 . . 3 (𝜑 → ∀𝑥𝜑)
2 hbralrimi.2 . . 3 (𝜑 → (𝑥𝐴𝜓))
31, 2alrimih 1823 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝜓))
4 df-ral 3146 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
53, 4sylibr 236 1 (𝜑 → ∀𝑥𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1534  wcel 2113  wral 3141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809
This theorem depends on definitions:  df-bi 209  df-ral 3146
This theorem is referenced by:  ralrimiv  3184  ralrimi  3219  bnj1145  32269
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