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Theorem ralim 2380
 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 2311 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
2 ax-2 6 . . . 4 ((𝑥𝐴 → (𝜑𝜓)) → ((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
32al2imi 1347 . . 3 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) → (∀𝑥(𝑥𝐴𝜑) → ∀𝑥(𝑥𝐴𝜓)))
41, 3sylbi 114 . 2 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥(𝑥𝐴𝜑) → ∀𝑥(𝑥𝐴𝜓)))
5 df-ral 2311 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
6 df-ral 2311 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
74, 5, 63imtr4g 194 1 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241   ∈ 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 This theorem depends on definitions:  df-bi 110  df-ral 2311 This theorem is referenced by:  ral2imi  2385  trint  3869  peano2  4318  mpteqb  5261  lbzbi  8551  r19.29uz  9590
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