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Theorem mof 2625
Description: Version of df-mo 2601 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 8-Mar-1995.) Extract dfmo 2660 from this proof, and prove mof 2625 from it (as of 30-Sep-2022, directly from df-mo 2601). (Revised by Wolf Lammen, 28-May-2019.) Avoid ax-13 2382. (Revised by Wolf Lammen, 16-Oct-2022.)
Hypothesis
Ref Expression
mof.1 𝑦𝜑
Assertion
Ref Expression
mof (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mof
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-mo 2601 . 2 (∃*𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 mof.1 . . . . 5 𝑦𝜑
3 nfv 1915 . . . . 5 𝑦 𝑥 = 𝑧
42, 3nfim 1897 . . . 4 𝑦(𝜑𝑥 = 𝑧)
54nfal 2334 . . 3 𝑦𝑥(𝜑𝑥 = 𝑧)
6 nfv 1915 . . 3 𝑧𝑥(𝜑𝑥 = 𝑦)
7 equequ2 2033 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
87imbi2d 344 . . . 4 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜑𝑥 = 𝑦)))
98albidv 1921 . . 3 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
105, 6, 9cbvexv1 2354 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
111, 10bitri 278 1 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  wex 1781  wnf 1785  ∃*wmo 2599
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  ax-6 1970  ax-7 2015  ax-10 2143  ax-11 2159  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-mo 2601
This theorem is referenced by:  mo3  2626  mo  2627  rmo2  3819  nmo  30265  bj-eu3f  34281  dffun3f  45205
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