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Theorem equvinv 1946
 Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2006, ax-13 2234. (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.)
Assertion
Ref Expression
equvinv (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvinv
StepHypRef Expression
1 ax6ev 1877 . . 3 𝑧 𝑧 = 𝑥
2 equtrr 1936 . . . . 5 (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))
32ancld 574 . . . 4 (𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝑧 = 𝑥𝑧 = 𝑦)))
43eximdv 1833 . . 3 (𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑥 → ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦)))
51, 4mpi 20 . 2 (𝑥 = 𝑦 → ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
6 ax7 1930 . . . 4 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
76imp 444 . . 3 ((𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)
87exlimiv 1845 . 2 (∃𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)
95, 8impbii 198 1 (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by:  equvelv  1950  ax8  1983  ax9  1990  ax13  2237
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