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Theorem ax13lem1 2424
Description: A version of ax13v 2423 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. The proof of ax13 2425 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 2129 . 2 (𝑧 = 𝑦 → ∃𝑤(𝑧 = 𝑤𝑦 = 𝑤))
2 ax13v 2423 . . . . 5 𝑥 = 𝑦 → (𝑦 = 𝑤 → ∀𝑥 𝑦 = 𝑤))
3 equeucl 2121 . . . . . 6 (𝑧 = 𝑤 → (𝑦 = 𝑤𝑧 = 𝑦))
43alimdv 2007 . . . . 5 (𝑧 = 𝑤 → (∀𝑥 𝑦 = 𝑤 → ∀𝑥 𝑧 = 𝑦))
52, 4syl9 77 . . . 4 𝑥 = 𝑦 → (𝑧 = 𝑤 → (𝑦 = 𝑤 → ∀𝑥 𝑧 = 𝑦)))
65impd 398 . . 3 𝑥 = 𝑦 → ((𝑧 = 𝑤𝑦 = 𝑤) → ∀𝑥 𝑧 = 𝑦))
76exlimdv 2024 . 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 384  wal 1635  wex 1859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-13 2422
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860
This theorem is referenced by:  ax13  2425  ax6e  2426  ax13lem2  2465  nfeqf2  2466  nfeqf2OLD  2467  wl-19.8eqv  33642  wl-19.2reqv  33643
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