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Theorem 19.12 2342
 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 2364 and r19.12sn 4650. (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 2151 . . 3 𝑦𝑦𝜑
21nfex 2339 . 2 𝑦𝑥𝑦𝜑
3 sp 2177 . . 3 (∀𝑦𝜑𝜑)
43eximi 1831 . 2 (∃𝑥𝑦𝜑 → ∃𝑥𝜑)
52, 4alrimi 2208 1 (∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1531  ∃wex 1776 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-11 2156  ax-12 2172 This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1777  df-nf 1781 This theorem is referenced by:  nfald  2343  bj-nfald  34423  pm11.61  40718
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