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Theorem reubiia 2796
 Description: Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 14-Nov-2004.)
Hypothesis
Ref Expression
reubiia.1 (x A → (φψ))
Assertion
Ref Expression
reubiia (∃!x A φ∃!x A ψ)

Proof of Theorem reubiia
StepHypRef Expression
1 reubiia.1 . . . 4 (x A → (φψ))
21pm5.32i 618 . . 3 ((x A φ) ↔ (x A ψ))
32eubii 2213 . 2 (∃!x(x A φ) ↔ ∃!x(x A ψ))
4 df-reu 2621 . 2 (∃!x A φ∃!x(x A φ))
5 df-reu 2621 . 2 (∃!x A ψ∃!x(x A ψ))
63, 4, 53bitr4i 268 1 (∃!x A φ∃!x A ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∈ 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-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208  df-reu 2621 This theorem is referenced by:  reubii  2797
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