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Theorem ralm 3435
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 2396 . . . . . 6 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
21imbi2i 225 . . . . 5 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ (∃𝑥 𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)))
3 19.38 1637 . . . . 5 ((∃𝑥 𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)) → ∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)))
42, 3sylbi 120 . . . 4 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) → ∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)))
5 pm2.43 53 . . . . 5 ((𝑥𝐴 → (𝑥𝐴𝜑)) → (𝑥𝐴𝜑))
65alimi 1414 . . . 4 (∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)) → ∀𝑥(𝑥𝐴𝜑))
74, 6syl 14 . . 3 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) → ∀𝑥(𝑥𝐴𝜑))
87, 1sylibr 133 . 2 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) → ∀𝑥𝐴 𝜑)
9 ax-1 6 . 2 (∀𝑥𝐴 𝜑 → (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑))
108, 9impbii 125 1 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1312  wex 1451  wcel 1463  wral 2391
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 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-ral 2396
This theorem is referenced by:  raaan  3437
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