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Theorem euf 1921
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 1919 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 euf.1 . . . . 5 (𝜑 → ∀𝑦𝜑)
3 ax-17 1435 . . . . 5 (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)
42, 3hbbi 1456 . . . 4 ((𝜑𝑥 = 𝑧) → ∀𝑦(𝜑𝑥 = 𝑧))
54hbal 1382 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑦𝑥(𝜑𝑥 = 𝑧))
6 ax-17 1435 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑧𝑥(𝜑𝑥 = 𝑦))
7 equequ2 1615 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
87bibi2d 225 . . . 4 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜑𝑥 = 𝑦)))
98albidv 1721 . . 3 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
105, 6, 9cbvexh 1654 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
111, 10bitri 177 1 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257  wex 1397  ∃!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-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-eu 1919
This theorem is referenced by:  eu1  1941  eumo0  1947
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