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Theorem equvinivOLD 1963
Description: The forward implication of equvinv 1961. Obsolete as of 11-Apr-2021. Use equvinv 1961 instead. (Contributed by Wolf Lammen, 11-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
equvinivOLD (𝑥 = 𝑦 → ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvinivOLD
StepHypRef Expression
1 ax6ev 1892 . 2 𝑧 𝑧 = 𝑥
2 equtrr 1951 . . . 4 (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))
32ancld 575 . . 3 (𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝑧 = 𝑥𝑧 = 𝑦)))
43eximdv 1848 . 2 (𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑥 → ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦)))
51, 4mpi 20 1 (𝑥 = 𝑦 → ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  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 1841  ax-6 1890  ax-7 1937
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by: (None)
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