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Theorem aeveq 2024
 Description: The antecedent ∀𝑥𝑥 = 𝑦 with a dv condition (typical of a one-object universe) forces equality of everything. (Contributed by Wolf Lammen, 19-Mar-2021.)
Assertion
Ref Expression
aeveq (∀𝑥 𝑥 = 𝑦𝑧 = 𝑡)
Distinct variable group:   𝑥,𝑦

Proof of Theorem aeveq
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 aevlem 2023 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑧)
2 ax6ev 1947 . . 3 𝑢 𝑢 = 𝑡
3 ax7 1989 . . . 4 (𝑢 = 𝑧 → (𝑢 = 𝑡𝑧 = 𝑡))
43aleximi 1799 . . 3 (∀𝑢 𝑢 = 𝑧 → (∃𝑢 𝑢 = 𝑡 → ∃𝑢 𝑧 = 𝑡))
52, 4mpi 20 . 2 (∀𝑢 𝑢 = 𝑧 → ∃𝑢 𝑧 = 𝑡)
6 ax5e 1881 . 2 (∃𝑢 𝑧 = 𝑡𝑧 = 𝑡)
71, 5, 63syl 18 1 (∀𝑥 𝑥 = 𝑦𝑧 = 𝑡)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1521  ∃wex 1744 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:  aev  2025  2ax6e  2478  aevdemo  27447  wl-spae  33436
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