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Theorem eubidv 2046
Description: Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)
Hypothesis
Ref Expression
eubidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
eubidv (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem eubidv
StepHypRef Expression
1 nfv 1539 . 2 𝑥𝜑
2 eubidv.1 . 2 (𝜑 → (𝜓𝜒))
31, 2eubid 2045 1 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  ∃!weu 2038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-eu 2041
This theorem is referenced by:  eubii  2047  eueq2dc  2925  eueq3dc  2926  reuhypd  4489  feu  5417  funfveu  5547  dff4im  5683  acexmid  5895  upxp  14232  dedekindicc  14571
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