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Theorem wl-19.2reqv 33623
 Description: Under the assumption ¬ 𝑥 = 𝑦 the reverse direction of 19.2 2072 is provable from Tarski's FOL and ax13v 2421 only. Note that in conjunction with 19.2 2072 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 2463 . 2 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))
2 ax13lem1 2422 . 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 1635  ∃wex 1859 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-13 2420 This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860 This theorem is referenced by: (None)
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