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Theorem raleqi 2628
 Description: Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypothesis
Ref Expression
raleq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
raleqi (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem raleqi
StepHypRef Expression
1 raleq1i.1 . 2 𝐴 = 𝐵
2 raleq 2624 . 2 (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))
31, 2ax-mp 5 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   = wceq 1331  ∀wral 2414 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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419 This theorem is referenced by:  ralrab2  2844  ralprg  3569  raltpg  3571  omsinds  4530  ralxp  4677  ralrnmpo  5878  fzprval  9855  fztpval  9856  seq3f1olemp  10268  zsumdc  11146  infssuzex  11631  2prm  11797  nninfsellemdc  13195  nninfsellemsuc  13197
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