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Theorem 19.38 1611
Description: Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
19.38 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))

Proof of Theorem 19.38
StepHypRef Expression
1 hbe1 1429 . . 3 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
2 hba1 1478 . . 3 (∀𝑥𝜓 → ∀𝑥𝑥𝜓)
31, 2hbim 1482 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
4 19.8a 1527 . . 3 (𝜑 → ∃𝑥𝜑)
5 ax-4 1445 . . 3 (∀𝑥𝜓𝜓)
64, 5imim12i 58 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → (𝜑𝜓))
73, 6alrimih 1403 1 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287  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-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  19.23t  1612  sbi2v  1820  mo3h  2001  rgenm  3380  ralm  3382
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