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Theorem ralbid 2436
 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 274 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
41, 3ralbida 2432 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  Ⅎwnf 1437   ∈ 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  ax-4 1488 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-ral 2422 This theorem is referenced by:  ralbidv  2438  sbcralt  2988  riota5f  5760  mkvprop  7038  lble  8727  ellimc3apf  12830  strcollnft  13346
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