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Theorem ralbii2 3086
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 1815 . 2 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐵𝜓))
3 df-ral 3059 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
4 df-ral 3059 . 2 (∀𝑥𝐵 𝜓 ↔ ∀𝑥(𝑥𝐵𝜓))
52, 3, 43bitr4i 303 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1534  wcel 2105  wral 3058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805
This theorem depends on definitions:  df-bi 207  df-ral 3059
This theorem is referenced by:  ralbiia  3088  ralcom3  3094  raleqbii  3341  ralrab  3701  raldifb  4158  raldifsni  4799  reusv2  5408  dfsup2  9481  iscard2  10013  acnnum  10089  dfac9  10174  dfacacn  10179  raluz2  12936  ralrp  13052  isprm4  16717  isdomn2OLD  20728  sdrgacs  20818  isnrm2  23381  ismbl  25574  ellimc3  25928  dchrelbas2  27295  h1dei  31578  iineq1i  36177  ixpeq1i  36181  ralin  38229  fnwe2lem2  43039
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