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Theorem reubida 2485
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 ((φ x A) → (ψχ))
Assertion
Ref Expression
reubida (φ → (∃!x A ψ∃!x A χ))

Proof of Theorem reubida
StepHypRef Expression
1 reubida.1 . . 3 xφ
2 reubida.2 . . . 4 ((φ x A) → (ψχ))
32pm5.32da 425 . . 3 (φ → ((x A ψ) ↔ (x A χ)))
41, 3eubid 1904 . 2 (φ → (∃!x(x A ψ) ↔ ∃!x(x A χ)))
5 df-reu 2307 . 2 (∃!x A ψ∃!x(x A ψ))
6 df-reu 2307 . 2 (∃!x A χ∃!x(x A χ))
74, 5, 63bitr4g 212 1 (φ → (∃!x A ψ∃!x A χ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wnf 1346   wcel 1390  ∃!weu 1897  ∃!wreu 2302
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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-eu 1900  df-reu 2307
This theorem is referenced by:  reubidva  2486
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