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Theorem ralbid 2468
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 2464 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wnf 1453  wcel 2141  wral 2448
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 1440  ax-gen 1442  ax-4 1503
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-ral 2453
This theorem is referenced by:  ralbidv  2470  sbcralt  3031  riota5f  5831  mkvprop  7131  lble  8852  ellimc3apf  13384  strcollnft  13981
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