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Theorem 19.12 2085
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 2124 and r19.12sn 4066. (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 2032 . . 3 𝑦𝑦𝜑
21nfex 2083 . 2 𝑦𝑥𝑦𝜑
3 sp 1990 . . 3 (∀𝑦𝜑𝜑)
43eximi 1738 . 2 (∃𝑥𝑦𝜑 → ∃𝑥𝜑)
52, 4alrimi 2008 1 (∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1466  wex 1692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-11 1970  ax-12 1983
This theorem depends on definitions:  df-bi 192  df-ex 1693  df-nf 1697
This theorem is referenced by:  pm11.61  37100
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