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Theorem equvinvOLD 2130
Description: Obsolete version of equvinv 2129 as of 12-Jul-2022. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2185, ax-13 2375. (Revised by Wolf Lammen, 10-Jun-2019.) Move the quantified variable (𝑧) to the left of the equality signs. (Revised by Wolf Lammen, 11-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
equvinvOLD (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvinvOLD
StepHypRef Expression
1 ax6ev 2074 . . 3 𝑧 𝑧 = 𝑥
2 equtrr 2121 . . . . 5 (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))
32ancld 547 . . . 4 (𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝑧 = 𝑥𝑧 = 𝑦)))
43eximdv 2013 . . 3 (𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑥 → ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦)))
51, 4mpi 20 . 2 (𝑥 = 𝑦 → ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
6 ax7 2115 . . . 4 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
76imp 396 . . 3 ((𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)
87exlimiv 2026 . 2 (∃𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)
95, 8impbii 201 1 (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385  wex 1875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876
This theorem is referenced by: (None)
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