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Theorem ralbid 2488
Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)
Hypotheses
Ref Expression
ralbid.1 𝑥𝜑
ralbid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
ralbid (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))

Proof of Theorem ralbid
StepHypRef Expression
1 ralbid.1 . 2 𝑥𝜑
2 ralbid.2 . . 3 (𝜑 → (𝜓𝜒))
32adantr 276 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
41, 3ralbida 2484 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wnf 1471  wcel 2160  wral 2468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-4 1521
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-ral 2473
This theorem is referenced by:  ralbidv  2490  sbcralt  3054  riota5f  5875  mkvprop  7185  lble  8933  ellimc3apf  14581  strcollnft  15189
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