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Theorem ralbidv2 2967
Description: Formula-building rule for restricted universal quantifier (deduction rule). (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 1836 . 2 (𝜑 → (∀𝑥(𝑥𝐴𝜓) ↔ ∀𝑥(𝑥𝐵𝜒)))
3 df-ral 2901 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
4 df-ral 2901 . 2 (∀𝑥𝐵 𝜒 ↔ ∀𝑥(𝑥𝐵𝜒))
52, 3, 43bitr4g 302 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wal 1473  wcel 1977  wral 2896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827
This theorem depends on definitions:  df-bi 196  df-ral 2901
This theorem is referenced by:  ralbidva  2968  ralss  3631  oneqmini  5693  ordunisuc2  6936  dfsmo2  7331  wemapsolem  8338  zorn2lem1  9201  raluz  11612  limsupgle  14056  ello12  14095  elo12  14106  lo1resb  14143  rlimresb  14144  o1resb  14145  isprm3  15234  isprm7  15258  ist1  20935  ist1-2  20961  hausdiag  21258  xkopt  21268  cnflf  21616  cnfcf  21656  metcnp  22156  caucfil  22889  mdegleb  23628  eulerpartlemgvv  29765  filnetlem4  31546  hoidmvle  39490  elbigo2  42144
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