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Theorem reubida 3403
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by Mario Carneiro, 19-Nov-2016.)
Hypotheses
Ref Expression
rmobida.1 𝑥𝜑
rmobida.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
reubida (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))

Proof of Theorem reubida
StepHypRef Expression
1 rmobida.1 . . 3 𝑥𝜑
2 rmobida.2 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
32pm5.32da 579 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3eubid 2581 . 2 (𝜑 → (∃!𝑥(𝑥𝐴𝜓) ↔ ∃!𝑥(𝑥𝐴𝜒)))
5 df-reu 3377 . 2 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
6 df-reu 3377 . 2 (∃!𝑥𝐴 𝜒 ↔ ∃!𝑥(𝑥𝐴𝜒))
74, 5, 63bitr4g 313 1 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wnf 1785  wcel 2106  ∃!weu 2562  ∃!wreu 3374
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 1913  ax-6 1971  ax-7 2011  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-nf 1786  df-mo 2534  df-eu 2563  df-reu 3377
This theorem is referenced by:  reuan  3890  poimirlem25  36816
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