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Theorem rspn0 4356
Description: Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018.) Avoid ax-10 2141, ax-12 2177. (Revised by GG, 28-Jun-2024.)
Assertion
Ref Expression
rspn0 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem rspn0
StepHypRef Expression
1 n0 4353 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 df-ral 3062 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
3 exim 1834 . . . 4 (∀𝑥(𝑥𝐴𝜑) → (∃𝑥 𝑥𝐴 → ∃𝑥𝜑))
4 ax5e 1912 . . . 4 (∃𝑥𝜑𝜑)
53, 4syl6com 37 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥(𝑥𝐴𝜑) → 𝜑))
62, 5biimtrid 242 . 2 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑𝜑))
71, 6sylbi 217 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538  wex 1779  wcel 2108  wne 2940  wral 3061  c0 4333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-ne 2941  df-ral 3062  df-dif 3954  df-nul 4334
This theorem is referenced by:  hashge2el2dif  14519  rmodislmodlem  20927  rmodislmod  20928  scmatf1  22537  fusgrregdegfi  29587  rusgr1vtxlem  29605  upgrewlkle2  29624  zarclsiin  33870  ralralimp  47290
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