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Theorem ax13lem2 2451
Description: Lemma for nfeqf2 2452. This lemma is equivalent to ax13v 2409 with one distinct variable constraint removed. (Contributed by Wolf Lammen, 8-Sep-2018.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Oct-2020.) (New usage is discouraged.)
Assertion
Ref Expression
ax13lem2 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))
Distinct variable group:   𝑥,𝑧

Proof of Theorem ax13lem2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax13lem1 2410 . . . 4 𝑥 = 𝑦 → (𝑤 = 𝑦 → ∀𝑥 𝑤 = 𝑦))
2 equeucl 2109 . . . . . 6 (𝑧 = 𝑦 → (𝑤 = 𝑦𝑧 = 𝑤))
32eximi 1910 . . . . 5 (∃𝑥 𝑧 = 𝑦 → ∃𝑥(𝑤 = 𝑦𝑧 = 𝑤))
4 19.36v 2072 . . . . 5 (∃𝑥(𝑤 = 𝑦𝑧 = 𝑤) ↔ (∀𝑥 𝑤 = 𝑦𝑧 = 𝑤))
53, 4sylib 208 . . . 4 (∃𝑥 𝑧 = 𝑦 → (∀𝑥 𝑤 = 𝑦𝑧 = 𝑤))
61, 5syl9 77 . . 3 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → (𝑤 = 𝑦𝑧 = 𝑤)))
76alrimdv 2009 . 2 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑤(𝑤 = 𝑦𝑧 = 𝑤)))
8 equequ2 2111 . . 3 (𝑤 = 𝑦 → (𝑧 = 𝑤𝑧 = 𝑦))
98equsalvw 2089 . 2 (∀𝑤(𝑤 = 𝑦𝑧 = 𝑤) ↔ 𝑧 = 𝑦)
107, 9syl6ib 241 1 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853
This theorem is referenced by:  nfeqf2  2452  nfeqf2OLD  2453  wl-speqv  33644  wl-19.2reqv  33646
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