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Theorem euim 2681
Description: Add unique existential quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Proof shortened by Wolf Lammen, 1-Oct-2023.)
Assertion
Ref Expression
euim ((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑))

Proof of Theorem euim
StepHypRef Expression
1 euimmo 2680 . 2 (∀𝑥(𝜑𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))
2 exmoeub 2643 . . 3 (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))
32biimpd 232 . 2 (∃𝑥𝜑 → (∃*𝑥𝜑 → ∃!𝑥𝜑))
41, 3sylan9r 512 1 ((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536  wex 1781  ∃*wmo 2599  ∃!weu 2631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2601  df-eu 2632
This theorem is referenced by:  2eu1  2715  2eu1v  2716  dfatcolem  43798
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