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Theorem raleqbi1dv 2530
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 2522 . 2 (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))
2 raleqd.1 . . 3 (𝐴 = 𝐵 → (𝜑𝜓))
32ralbidv 2343 . 2 (𝐴 = 𝐵 → (∀𝑥𝐵 𝜑 ↔ ∀𝑥𝐵 𝜓))
41, 3bitrd 181 1 (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328
This theorem is referenced by:  frforeq2  4110  weeq2  4122  peano5  4349  isoeq4  5472  pitonn  6982  peano1nnnn  6986  peano2nnnn  6987  peano5nnnn  7024  peano5nni  7993  1nn  8001  peano2nn  8002  dfuzi  8407  bj-indeq  10440  bj-nntrans  10463
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