Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  raleq GIF version

Theorem raleq 2505
 Description: Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
Assertion
Ref Expression
raleq (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem raleq
StepHypRef Expression
1 nfcv 2178 . 2 𝑥𝐴
2 nfcv 2178 . 2 𝑥𝐵
31, 2raleqf 2501 1 (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243  ∀wral 2306 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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311 This theorem is referenced by:  raleqi  2509  raleqdv  2511  raleqbi1dv  2513  sbralie  2546  inteq  3618  iineq1  3671  bnd2  3926  frforeq2  4082  weeq2  4094  ordeq  4109  reg2exmid  4261  reg3exmid  4304  fncnv  4965  funimaexglem  4982  isoeq4  5444  acexmidlemv  5510  tfrlem1  5923  tfr0  5937  tfrlemisucaccv  5939  tfrlemi1  5946  tfrlemi14d  5947  tfrexlem  5948  ac6sfi  6352  rexanuz  9587  setindis  10092  bdsetindis  10094  strcoll2  10108
 Copyright terms: Public domain W3C validator