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Theorem raleqbi1dv 2634
Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
Hypothesis
Ref Expression
raleqd.1 (𝐴 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
raleqbi1dv (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem raleqbi1dv
StepHypRef Expression
1 raleq 2626 . 2 (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))
2 raleqd.1 . . 3 (𝐴 = 𝐵 → (𝜑𝜓))
32ralbidv 2437 . 2 (𝐴 = 𝐵 → (∀𝑥𝐵 𝜑 ↔ ∀𝑥𝐵 𝜓))
41, 3bitrd 187 1 (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wral 2416
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 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421
This theorem is referenced by:  frforeq2  4267  weeq2  4279  peano5  4512  isoeq4  5705  exmidomni  7014  pitonn  7670  peano1nnnn  7674  peano2nnnn  7675  peano5nnnn  7714  peano5nni  8737  1nn  8745  peano2nn  8746  dfuzi  9175  istopg  12182  isbasisg  12227  basis2  12231  eltg2  12238  ispsmet  12508  ismet  12529  isxmet  12530  metrest  12691  cncfval  12744  bj-indeq  13234  bj-nntrans  13256
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