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Theorem equvinv 2003
Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2059, ax-13 2282. (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 1947 . . 3 𝑧 𝑧 = 𝑥
2 equtrr 1995 . . . . 5 (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))
32ancld 575 . . . 4 (𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝑧 = 𝑥𝑧 = 𝑦)))
43eximdv 1886 . . 3 (𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑥 → ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦)))
51, 4mpi 20 . 2 (𝑥 = 𝑦 → ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
6 ax7 1989 . . . 4 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
76imp 444 . . 3 ((𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)
87exlimiv 1898 . 2 (∃𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)
95, 8impbii 199 1 (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  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:  equvelv  2005  ax8  2036  ax9  2043  ax13  2285  wl-ax8clv1  33508  wl-ax8clv2  33511  cossid  34370
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