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Theorem euf 1982
Description: A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)
Hypothesis
Ref Expression
euf.1 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
euf (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem euf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-eu 1980 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 euf.1 . . . . 5 (𝜑 → ∀𝑦𝜑)
3 ax-17 1491 . . . . 5 (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)
42, 3hbbi 1512 . . . 4 ((𝜑𝑥 = 𝑧) → ∀𝑦(𝜑𝑥 = 𝑧))
54hbal 1438 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑦𝑥(𝜑𝑥 = 𝑧))
6 ax-17 1491 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑧𝑥(𝜑𝑥 = 𝑦))
7 equequ2 1674 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
87bibi2d 231 . . . 4 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜑𝑥 = 𝑦)))
98albidv 1780 . . 3 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
105, 6, 9cbvexh 1713 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
111, 10bitri 183 1 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1314  wex 1453  ∃!weu 1977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-eu 1980
This theorem is referenced by:  eu1  2002  eumo0  2008
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