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Theorem excomim 1598
Description: One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
excomim (∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑)

Proof of Theorem excomim
StepHypRef Expression
1 19.8a 1527 . . 3 (𝜑 → ∃𝑥𝜑)
212eximi 1537 . 2 (∃𝑥𝑦𝜑 → ∃𝑥𝑦𝑥𝜑)
3 hbe1 1429 . . . 4 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
43hbex 1572 . . 3 (∃𝑦𝑥𝜑 → ∀𝑥𝑦𝑥𝜑)
5419.9h 1579 . 2 (∃𝑥𝑦𝑥𝜑 ↔ ∃𝑦𝑥𝜑)
62, 5sylib 120 1 (∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1426
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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  excom  1599  2euswapdc  2039
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