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Theorem reubiia 2488
 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 427 . . 3 ((x A φ) ↔ (x A ψ))
32eubii 1906 . 2 (∃!x(x A φ) ↔ ∃!x(x A ψ))
4 df-reu 2307 . 2 (∃!x A φ∃!x(x A φ))
5 df-reu 2307 . 2 (∃!x A ψ∃!x(x A ψ))
63, 4, 53bitr4i 201 1 (∃!x A φ∃!x A ψ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ 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-tru 1245  df-nf 1347  df-eu 1900  df-reu 2307 This theorem is referenced by:  reubii  2489
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