MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralbidv2 Structured version   Visualization version   GIF version

Theorem ralbidv2 3173
Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Apr-1997.)
Hypothesis
Ref Expression
ralbidv2.1 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
Assertion
Ref Expression
ralbidv2 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem ralbidv2
StepHypRef Expression
1 ralbidv2.1 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
21albidv 1923 . 2 (𝜑 → (∀𝑥(𝑥𝐴𝜓) ↔ ∀𝑥(𝑥𝐵𝜒)))
3 df-ral 3062 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
4 df-ral 3062 . 2 (∀𝑥𝐵 𝜒 ↔ ∀𝑥(𝑥𝐵𝜒))
52, 3, 43bitr4g 313 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539  wcel 2106  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913
This theorem depends on definitions:  df-bi 206  df-ral 3062
This theorem is referenced by:  ralbidva  3175  raleqbidv  3342  ralss  4054  oneqmini  6416  ordunisuc2  7835  dfsmo2  8349  wemapsolem  9547  zorn2lem1  10493  raluz  12882  limsupgle  15423  ello12  15462  elo12  15473  lo1resb  15510  rlimresb  15511  o1resb  15512  isprm3  16622  isprm7  16647  ist1-2  22858  hausdiag  23156  xkopt  23166  cnflf  23513  cnfcf  23553  metcnp  24057  caucfil  24807  mdegleb  25589  islinds5  32525  islbs5  32541  eulerpartlemgvv  33444  filnetlem4  35352  mnuunid  43118  hoidmvle  45395  elbigo2  47316  ralbidb  47563  ralbidc  47564
  Copyright terms: Public domain W3C validator