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Theorem reubida 2548
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 440 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3eubid 1955 . 2 (𝜑 → (∃!𝑥(𝑥𝐴𝜓) ↔ ∃!𝑥(𝑥𝐴𝜒)))
5 df-reu 2366 . 2 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
6 df-reu 2366 . 2 (∃!𝑥𝐴 𝜒 ↔ ∃!𝑥(𝑥𝐴𝜒))
74, 5, 63bitr4g 221 1 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wnf 1394  wcel 1438  ∃!weu 1948  ∃!wreu 2361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-eu 1951  df-reu 2366
This theorem is referenced by:  reubidva  2549
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