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Theorem ralbiia 2392
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 178 . 2 ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓))
32ralbii2 2388 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wcel 1438  wral 2359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383
This theorem depends on definitions:  df-bi 115  df-ral 2364
This theorem is referenced by:  frind  4170  poinxp  4495  soinxp  4496  seinxp  4497  dffun8  5029  funcnv3  5062  fncnv  5066  fnres  5116  fvreseq  5387  isoini2  5580  smores  6039  finomni  6775  caucvgre  10379
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