MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralbii2 Structured version   Visualization version   GIF version

Theorem ralbii2 3078
Description: Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005.)
Hypothesis
Ref Expression
ralbii2.1 ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜓))
Assertion
Ref Expression
ralbii2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓)

Proof of Theorem ralbii2
StepHypRef Expression
1 ralbii2.1 . . 3 ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜓))
21albii 1813 . 2 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐵𝜓))
3 df-ral 3051 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
4 df-ral 3051 . 2 (∀𝑥𝐵 𝜓 ↔ ∀𝑥(𝑥𝐵𝜓))
52, 3, 43bitr4i 302 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531  wcel 2098  wral 3050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803
This theorem depends on definitions:  df-bi 206  df-ral 3051
This theorem is referenced by:  ralbiia  3080  ralcom3  3086  raleqbii  3327  ralrab  3685  raldifb  4141  raldifsni  4800  reusv2  5403  dfsup2  9474  iscard2  10006  acnnum  10082  dfac9  10166  dfacacn  10171  raluz2  12919  ralrp  13034  isprm4  16671  sdrgacs  20718  isdomn2  21280  isnrm2  23323  ismbl  25516  ellimc3  25869  dchrelbas2  27235  h1dei  31452  ralin  37870  fnwe2lem2  42622
  Copyright terms: Public domain W3C validator