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Theorem ralim 2491
Description: Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)
Assertion
Ref Expression
ralim (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓))

Proof of Theorem ralim
StepHypRef Expression
1 df-ral 2421 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
2 ax-2 7 . . . 4 ((𝑥𝐴 → (𝜑𝜓)) → ((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
32al2imi 1434 . . 3 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) → (∀𝑥(𝑥𝐴𝜑) → ∀𝑥(𝑥𝐴𝜓)))
41, 3sylbi 120 . 2 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥(𝑥𝐴𝜑) → ∀𝑥(𝑥𝐴𝜓)))
5 df-ral 2421 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
6 df-ral 2421 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
74, 5, 63imtr4g 204 1 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1329  wcel 1480  wral 2416
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 1423  ax-gen 1425
This theorem depends on definitions:  df-bi 116  df-ral 2421
This theorem is referenced by:  ral2imi  2497  trint  4041  peano2  4509  mpteqb  5511  mptelixpg  6628  lbzbi  9420  r19.29uz  10776  alzdvds  11563
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