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Theorem eubii 1957
Description: Introduce unique existential 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 1956 . 2 (⊤ → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓))
43mptru 1298 1 (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)
Colors of variables: wff set class
Syntax hints:  wb 103  wtru 1290  ∃!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-eu 1951
This theorem is referenced by:  cbveu  1972  2eu7  2042  reubiia  2551  cbvreu  2588  reuv  2638  euxfr2dc  2798  euxfrdc  2799  2reuswapdc  2817  reuun2  3280  zfnuleu  3955  copsexg  4062  funeu2  5027  funcnv3  5062  fneu2  5105  tz6.12  5316  f1ompt  5434  fsn  5453  climreu  10649  divalgb  11018
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