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Theorem euf 1953
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 1951 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 euf.1 . . . . 5 (𝜑 → ∀𝑦𝜑)
3 ax-17 1464 . . . . 5 (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)
42, 3hbbi 1485 . . . 4 ((𝜑𝑥 = 𝑧) → ∀𝑦(𝜑𝑥 = 𝑧))
54hbal 1411 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑦𝑥(𝜑𝑥 = 𝑧))
6 ax-17 1464 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑧𝑥(𝜑𝑥 = 𝑦))
7 equequ2 1646 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
87bibi2d 230 . . . 4 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜑𝑥 = 𝑦)))
98albidv 1752 . . 3 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
105, 6, 9cbvexh 1685 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
111, 10bitri 182 1 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287  wex 1426  ∃!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-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-eu 1951
This theorem is referenced by:  eu1  1973  eumo0  1979
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