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Theorem equvinva 2129
Description: A modified version of the forward implication of equvinv 2127 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 2112 . 2 𝑧 𝑦 = 𝑧
2 equtr 2118 . . . 4 (𝑥 = 𝑦 → (𝑦 = 𝑧𝑥 = 𝑧))
32ancrd 547 . . 3 (𝑥 = 𝑦 → (𝑦 = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)))
43eximdv 2012 . 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 384  wex 1874
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 2070  ax-7 2105
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875
This theorem is referenced by:  ax13lem1  2354  nfeqf  2401  bj-ssbequ2  33100  wl-ax13lem1  33743
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