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Theorem reubida 2491
 Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by Mario Carneiro, 19-Nov-2016.)
Hypotheses
Ref Expression
reubida.1 𝑥𝜑
reubida.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
reubida (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))

Proof of Theorem reubida
StepHypRef Expression
1 reubida.1 . . 3 𝑥𝜑
2 reubida.2 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
32pm5.32da 425 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3eubid 1907 . 2 (𝜑 → (∃!𝑥(𝑥𝐴𝜓) ↔ ∃!𝑥(𝑥𝐴𝜒)))
5 df-reu 2313 . 2 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
6 df-reu 2313 . 2 (∃!𝑥𝐴 𝜒 ↔ ∃!𝑥(𝑥𝐴𝜒))
74, 5, 63bitr4g 212 1 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  Ⅎwnf 1349   ∈ wcel 1393  ∃!weu 1900  ∃!wreu 2308 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-eu 1903  df-reu 2313 This theorem is referenced by:  reubidva  2492
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