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Theorem wl-19.2reqv 33641
 Description: Under the assumption ¬ 𝑥 = 𝑦 the reverse direction of 19.2 2058 is provable from Tarski's FOL and ax13v 2392 only. Note that in conjunction with 19.2 2058 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 2441 . 2 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))
2 ax13lem1 2393 . 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 1630  ∃wex 1853 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-13 2391 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854 This theorem is referenced by: (None)
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