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Theorem eubii 1925
Description: Introduce uniqueness quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypothesis
Ref Expression
eubii.1 (𝜑𝜓)
Assertion
Ref Expression
eubii (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)

Proof of Theorem eubii
StepHypRef Expression
1 eubii.1 . . . 4 (𝜑𝜓)
21a1i 9 . . 3 (⊤ → (𝜑𝜓))
32eubidv 1924 . 2 (⊤ → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓))
43trud 1268 1 (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)
Colors of variables: wff set class
Syntax hints:  wb 102  wtru 1260  ∃!weu 1916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-eu 1919
This theorem is referenced by:  cbveu  1940  2eu7  2010  reubiia  2511  cbvreu  2548  reuv  2590  euxfr2dc  2749  euxfrdc  2750  2reuswapdc  2766  reuun2  3248  zfnuleu  3909  copsexg  4009  funeu2  4955  funcnv3  4989  fneu2  5032  tz6.12  5229  f1ompt  5348  fsn  5363  climreu  10049  divalgb  10237
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