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Theorem ralm 3304
 Description: Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)
Assertion
Ref Expression
ralm ((x x Ax A φ) ↔ x A φ)

Proof of Theorem ralm
StepHypRef Expression
1 df-ral 2289 . . . . . 6 (x A φx(x Aφ))
21imbi2i 215 . . . . 5 ((x x Ax A φ) ↔ (x x Ax(x Aφ)))
3 19.38 1548 . . . . 5 ((x x Ax(x Aφ)) → x(x A → (x Aφ)))
42, 3sylbi 114 . . . 4 ((x x Ax A φ) → x(x A → (x Aφ)))
5 pm2.43 47 . . . . 5 ((x A → (x Aφ)) → (x Aφ))
65alimi 1324 . . . 4 (x(x A → (x Aφ)) → x(x Aφ))
74, 6syl 14 . . 3 ((x x Ax A φ) → x(x Aφ))
87, 1sylibr 137 . 2 ((x x Ax A φ) → x A φ)
9 ax-1 5 . 2 (x A φ → (x x Ax A φ))
108, 9impbii 117 1 ((x x Ax A φ) ↔ x A φ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1226  ∃wex 1362   ∈ wcel 1374  ∀wral 2284 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 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-ial 1409  ax-i5r 1410 This theorem depends on definitions:  df-bi 110  df-ral 2289 This theorem is referenced by:  raaan  3306
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