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Theorem equvinv 1956
Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2016, ax-13 2245. (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 1887 . . 3 𝑧 𝑧 = 𝑥
2 equtrr 1946 . . . . 5 (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))
32ancld 575 . . . 4 (𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝑧 = 𝑥𝑧 = 𝑦)))
43eximdv 1843 . . 3 (𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑥 → ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦)))
51, 4mpi 20 . 2 (𝑥 = 𝑦 → ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
6 ax7 1940 . . . 4 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
76imp 445 . . 3 ((𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)
87exlimiv 1855 . 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 384  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by:  equvelv  1960  ax8  1993  ax9  2000  ax13  2248  wl-ax8clv1  33010  wl-ax8clv2  33013
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