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Theorem ralbiia 2452
Description: Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)
Hypothesis
Ref Expression
ralbiia.1 (𝑥𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ralbiia (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓)

Proof of Theorem ralbiia
StepHypRef Expression
1 ralbiia.1 . . 3 (𝑥𝐴 → (𝜑𝜓))
21pm5.74i 179 . 2 ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓))
32ralbii2 2448 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 1481  wral 2417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426
This theorem depends on definitions:  df-bi 116  df-ral 2422
This theorem is referenced by:  frind  4281  poinxp  4615  soinxp  4616  seinxp  4617  dffun8  5158  funcnv3  5192  fncnv  5196  fnres  5246  fvreseq  5531  isoini2  5727  smores  6196  resixp  6634  finomni  7019  caucvgre  10784
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