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Theorem 19.12 2162
Description: Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! But sometimes the converse does hold, as in 19.12vv 2178 and r19.12sn 4246. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
Assertion
Ref Expression
19.12 (∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Proof of Theorem 19.12
StepHypRef Expression
1 nfa1 2026 . . 3 𝑦𝑦𝜑
21nfex 2152 . 2 𝑦𝑥𝑦𝜑
3 sp 2051 . . 3 (∀𝑦𝜑𝜑)
43eximi 1760 . 2 (∃𝑥𝑦𝜑 → ∃𝑥𝜑)
52, 4alrimi 2080 1 (∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  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-10 2017  ax-11 2032  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1703  df-nf 1708
This theorem is referenced by:  nfald  2163  pm11.61  38413
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