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Theorem equvinvOLD 1949
 Description: Obsolete version of equvinv 1946 as of 11-Apr-2021. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2006, ax-13 2234. (Revised by Wolf Lammen, 10-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
equvinvOLD (𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧𝑦 = 𝑧))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvinvOLD
StepHypRef Expression
1 equviniva 1947 . 2 (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑦 = 𝑧))
2 equtrr 1936 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
32equcoms 1934 . . . 4 (𝑦 = 𝑧 → (𝑥 = 𝑧𝑥 = 𝑦))
43impcom 445 . . 3 ((𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)
54exlimiv 1845 . 2 (∃𝑧(𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)
61, 5impbii 198 1 (𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧𝑦 = 𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ 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: (None)
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