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Theorem equvinva 2030
Description: A modified version of the forward implication of equvinv 2029 adapted to common usage. (Contributed by Wolf Lammen, 8-Sep-2018.)
Assertion
Ref Expression
equvinva (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑦 = 𝑧))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvinva
StepHypRef Expression
1 ax6evr 2015 . 2 𝑧 𝑦 = 𝑧
2 equtr 2021 . . . 4 (𝑥 = 𝑦 → (𝑦 = 𝑧𝑥 = 𝑧))
32ancrd 554 . . 3 (𝑥 = 𝑦 → (𝑦 = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)))
43eximdv 1911 . 2 (𝑥 = 𝑦 → (∃𝑧 𝑦 = 𝑧 → ∃𝑧(𝑥 = 𝑧𝑦 = 𝑧)))
51, 4mpi 20 1 (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1774
This theorem is referenced by:  sbequ2  2242  sbequ2OLD  2243  ax13lem1  2385  nfeqf  2392  wl-ax13lem1  34733
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