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Theorem aevOLD 2159
 Description: Obsolete proof of aev 1980 as of 19-Mar-2021. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 2031. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2245, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
aevOLD (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)
Distinct variable group:   𝑥,𝑦

Proof of Theorem aevOLD
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 aevlem 1978 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑤)
2 ax6ev 1887 . . . 4 𝑢 𝑢 = 𝑣
3 ax7 1940 . . . . 5 (𝑢 = 𝑤 → (𝑢 = 𝑣𝑤 = 𝑣))
43aleximi 1756 . . . 4 (∀𝑢 𝑢 = 𝑤 → (∃𝑢 𝑢 = 𝑣 → ∃𝑢 𝑤 = 𝑣))
52, 4mpi 20 . . 3 (∀𝑢 𝑢 = 𝑤 → ∃𝑢 𝑤 = 𝑣)
6 ax5e 1838 . . 3 (∃𝑢 𝑤 = 𝑣𝑤 = 𝑣)
71, 5, 63syl 18 . 2 (∀𝑥 𝑥 = 𝑦𝑤 = 𝑣)
8 axc16g 2130 . 2 (∀𝑥 𝑥 = 𝑦 → (𝑤 = 𝑣 → ∀𝑧 𝑤 = 𝑣))
97, 8mpd 15 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1478  ∃wex 1701 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702 This theorem is referenced by: (None)
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