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Theorem ralbiia 2381
 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 2377 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 103   ∈ wcel 1434  ∀wral 2349 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 1377  ax-gen 1379 This theorem depends on definitions:  df-bi 115  df-ral 2354 This theorem is referenced by:  frind  4115  poinxp  4435  soinxp  4436  seinxp  4437  dffun8  4959  funcnv3  4992  fncnv  4996  fnres  5046  fvreseq  5303  isoini2  5489  smores  5941  caucvgre  10005
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