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Theorem reubida 2793
 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 622 . . 3 (φ → ((x A ψ) ↔ (x A χ)))
41, 3eubid 2211 . 2 (φ → (∃!x(x A ψ) ↔ ∃!x(x A χ)))
5 df-reu 2621 . 2 (∃!x A ψ∃!x(x A ψ))
6 df-reu 2621 . 2 (∃!x A χ∃!x(x A χ))
74, 5, 63bitr4g 279 1 (φ → (∃!x A ψ∃!x A χ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   ∈ wcel 1710  ∃!weu 2204  ∃!wreu 2616 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-eu 2208  df-reu 2621 This theorem is referenced by:  reubidva  2794
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