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Theorem ralrimi 2390
 Description: Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.)
Hypotheses
Ref Expression
ralrimi.1 𝑥𝜑
ralrimi.2 (𝜑 → (𝑥𝐴𝜓))
Assertion
Ref Expression
ralrimi (𝜑 → ∀𝑥𝐴 𝜓)

Proof of Theorem ralrimi
StepHypRef Expression
1 ralrimi.1 . . 3 𝑥𝜑
2 ralrimi.2 . . 3 (𝜑 → (𝑥𝐴𝜓))
31, 2alrimi 1415 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝜓))
4 df-ral 2311 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
53, 4sylibr 137 1 (𝜑 → ∀𝑥𝐴 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241  Ⅎwnf 1349   ∈ wcel 1393  ∀wral 2306 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311 This theorem is referenced by:  ralrimiv  2391  reximdai  2417  r19.12  2422  rexlimd  2430  rexlimd2  2431  r19.29af2  2452  r19.37  2462  ralidm  3321  iineq2d  3677  mpteq2da  3846  onintonm  4243  mpteqb  5261  eusvobj2  5498  tfri3  5953
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