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Theorem 19.12 1665
Description: Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
19.12 (∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Proof of Theorem 19.12
StepHypRef Expression
1 hba1 1540 . . 3 (∀𝑦𝜑 → ∀𝑦𝑦𝜑)
21hbex 1636 . 2 (∃𝑥𝑦𝜑 → ∀𝑦𝑥𝑦𝜑)
3 ax-4 1510 . . 3 (∀𝑦𝜑𝜑)
43eximi 1600 . 2 (∃𝑥𝑦𝜑 → ∃𝑥𝜑)
52, 4alrimih 1469 1 (∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1351  wex 1492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  hbexd  1694  nfexd  1761  cbvexdh  1926
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