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Theorem reubida 3341
 Description: Formula-building rule for restricted existential quantifier (deduction form). (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 582 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3eubid 2648 . 2 (𝜑 → (∃!𝑥(𝑥𝐴𝜓) ↔ ∃!𝑥(𝑥𝐴𝜒)))
5 df-reu 3113 . 2 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
6 df-reu 3113 . 2 (∃!𝑥𝐴 𝜒 ↔ ∃!𝑥(𝑥𝐴𝜒))
74, 5, 63bitr4g 317 1 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  Ⅎwnf 1785   ∈ wcel 2111  ∃!weu 2628  ∃!wreu 3108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-mo 2598  df-eu 2629  df-reu 3113 This theorem is referenced by:  reuan  3827  poimirlem25  35233
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