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Theorem aev 2150
Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 2198. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2352, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 2211. (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 2148 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑣 𝑣 = 𝑤)
2 aeveq 2149 . . 3 (∀𝑣 𝑣 = 𝑤𝑡 = 𝑢)
32alrimiv 2022 . 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 1650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875
This theorem is referenced by:  aev2  2153  aev2ALT  2154  axc11n  2406  axc16gALT  2458  aevdemo  27711  axc11n11r  33041  wl-naev  33661  wl-hbnaev  33664  wl-ax11-lem2  33720
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