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Theorem aev 2025
 Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 2074. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2282, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 2087. (Revised by Wolf Lammen, 19-Mar-2021.)
Assertion
Ref Expression
aev (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑡 = 𝑢)
Distinct variable group:   𝑥,𝑦

Proof of Theorem aev
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 aevlem 2023 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑣 𝑣 = 𝑤)
2 aeveq 2024 . . 3 (∀𝑣 𝑣 = 𝑤𝑡 = 𝑢)
32alrimiv 1895 . 2 (∀𝑣 𝑣 = 𝑤 → ∀𝑧 𝑡 = 𝑢)
41, 3syl 17 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑡 = 𝑢)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745 This theorem is referenced by:  aev2  2028  aev2ALT  2029  axc16nfOLD  2199  axc11n  2342  axc11nOLD  2343  axc16gALT  2395  aevdemo  27447  axc11n11r  32798  wl-naev  33432  wl-hbnaev  33435  wl-ax11-lem2  33493
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