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Theorem ax13lem1 2394
Description: A version of ax13v 2393 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. The proof of ax13 2395 bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)
Assertion
Ref Expression
ax13lem1 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Distinct variable group:   𝑥,𝑧

Proof of Theorem ax13lem1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 equvinva 2038 . 2 (𝑧 = 𝑦 → ∃𝑤(𝑧 = 𝑤𝑦 = 𝑤))
2 ax13v 2393 . . . . 5 𝑥 = 𝑦 → (𝑦 = 𝑤 → ∀𝑥 𝑦 = 𝑤))
3 equeucl 2032 . . . . . 6 (𝑧 = 𝑤 → (𝑦 = 𝑤𝑧 = 𝑦))
43alimdv 1918 . . . . 5 (𝑧 = 𝑤 → (∀𝑥 𝑦 = 𝑤 → ∀𝑥 𝑧 = 𝑦))
52, 4syl9 77 . . . 4 𝑥 = 𝑦 → (𝑧 = 𝑤 → (𝑦 = 𝑤 → ∀𝑥 𝑧 = 𝑦)))
65impd 414 . . 3 𝑥 = 𝑦 → ((𝑧 = 𝑤𝑦 = 𝑤) → ∀𝑥 𝑧 = 𝑦))
76exlimdv 1935 . 2 𝑥 = 𝑦 → (∃𝑤(𝑧 = 𝑤𝑦 = 𝑤) → ∀𝑥 𝑧 = 𝑦))
81, 7syl5 34 1 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wal 1536  wex 1781
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-13 2392
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by:  ax13  2395  ax13lem2  2396  nfeqf2  2397  ax6e  2403  wl-19.8eqv  34828  wl-19.2reqv  34829
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