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Theorem aev 1969
Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 2020. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2229, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 2032. (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 1967 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑣 𝑣 = 𝑤)
2 aeveq 1968 . . 3 (∀𝑣 𝑣 = 𝑤𝑡 = 𝑢)
32alrimiv 1841 . 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 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by:  aev2  1972  aev2ALT  1973  axc16nfOLD  2145  axc11n  2291  axc11nOLD  2292  axc16gALT  2351  aevdemo  26472  axc11n11r  31663  wl-naev  32281  wl-hbnaev  32284  wl-ax11-lem2  32342
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