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Theorem 19.12 1643
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 1520 . . 3 (∀𝑦𝜑 → ∀𝑦𝑦𝜑)
21hbex 1615 . 2 (∃𝑥𝑦𝜑 → ∀𝑦𝑥𝑦𝜑)
3 ax-4 1487 . . 3 (∀𝑦𝜑𝜑)
43eximi 1579 . 2 (∃𝑥𝑦𝜑 → ∃𝑥𝜑)
52, 4alrimih 1445 1 (∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1329  wex 1468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  hbexd  1672  nfexd  1734  cbvexdh  1896
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