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Theorem reubidv 2653
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 17-Oct-1996.)
Hypothesis
Ref Expression
reubidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
reubidv (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem reubidv
StepHypRef Expression
1 reubidv.1 . . 3 (𝜑 → (𝜓𝜒))
21adantr 274 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
32reubidva 2652 1 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 2141  ∃!wreu 2450
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-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-eu 2022  df-reu 2455
This theorem is referenced by:  reueqd  2675  sbcreug  3035  xpf1o  6820  srpospr  7738  creur  8868  creui  8869  divalg2  11878
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