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Theorem ralm 3382
Description: Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)
Assertion
Ref Expression
ralm ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ ∀𝑥𝐴 𝜑)

Proof of Theorem ralm
StepHypRef Expression
1 df-ral 2364 . . . . . 6 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
21imbi2i 224 . . . . 5 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ (∃𝑥 𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)))
3 19.38 1611 . . . . 5 ((∃𝑥 𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)) → ∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)))
42, 3sylbi 119 . . . 4 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) → ∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)))
5 pm2.43 52 . . . . 5 ((𝑥𝐴 → (𝑥𝐴𝜑)) → (𝑥𝐴𝜑))
65alimi 1389 . . . 4 (∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)) → ∀𝑥(𝑥𝐴𝜑))
74, 6syl 14 . . 3 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) → ∀𝑥(𝑥𝐴𝜑))
87, 1sylibr 132 . 2 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) → ∀𝑥𝐴 𝜑)
9 ax-1 5 . 2 (∀𝑥𝐴 𝜑 → (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑))
108, 9impbii 124 1 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287  wex 1426  wcel 1438  wral 2359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-ral 2364
This theorem is referenced by:  raaan  3384
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