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Theorem 19.12 1598
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 1476 . . 3 (∀𝑦𝜑 → ∀𝑦𝑦𝜑)
21hbex 1570 . 2 (∃𝑥𝑦𝜑 → ∀𝑦𝑥𝑦𝜑)
3 ax-4 1443 . . 3 (∀𝑦𝜑𝜑)
43eximi 1534 . 2 (∃𝑥𝑦𝜑 → ∃𝑥𝜑)
52, 4alrimih 1401 1 (∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1285  wex 1424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  hbexd  1627  nfexd  1688  cbvexdh  1846
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