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Theorem euf 2662
Description: Version of eu6 2660 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2018.) Avoid ax-13 2392. (Revised by Wolf Lammen, 16-Oct-2022.)
Hypothesis
Ref Expression
euf.1 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 eu6 2660 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 euf.1 . . . . 5 𝑦𝜑
3 nfv 1916 . . . . 5 𝑦 𝑥 = 𝑧
42, 3nfbi 1905 . . . 4 𝑦(𝜑𝑥 = 𝑧)
54nfal 2344 . . 3 𝑦𝑥(𝜑𝑥 = 𝑧)
6 nfv 1916 . . 3 𝑧𝑥(𝜑𝑥 = 𝑦)
7 equequ2 2034 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
87bibi2d 346 . . . 4 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜑𝑥 = 𝑦)))
98albidv 1922 . . 3 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
105, 6, 9cbvexv1 2364 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
111, 10bitri 278 1 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1536  wex 1781  wnf 1785  ∃!weu 2654
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 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-11 2162  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-mo 2624  df-eu 2655
This theorem is referenced by:  eu1  2697
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