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Theorem wl-19.2reqv 33281
 Description: Under the assumption ¬ 𝑥 = 𝑦 the reverse direction of 19.2 1890 is provable from Tarski's FOL and ax13v 2245 only. Note that in conjunction with 19.2 1890 in fact (¬ 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦 ↔ ∃𝑥𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
Assertion
Ref Expression
wl-19.2reqv 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Distinct variable group:   𝑥,𝑧

Proof of Theorem wl-19.2reqv
StepHypRef Expression
1 ax13lem2 2294 . 2 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))
2 ax13lem1 2246 . 2 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
31, 2syld 47 1 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1479  ∃wex 1702 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-13 2244 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703 This theorem is referenced by: (None)
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