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Theorem 19.38 1566
 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 1384 . . 3 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
2 hba1 1433 . . 3 (∀𝑥𝜓 → ∀𝑥𝑥𝜓)
31, 2hbim 1437 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
4 19.8a 1482 . . 3 (𝜑 → ∃𝑥𝜑)
5 ax-4 1400 . . 3 (∀𝑥𝜓𝜓)
64, 5imim12i 53 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → (𝜑𝜓))
73, 6alrimih 1358 1 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  19.23t  1567  sbi2v  1772  mo3h  1953  rgenm  3323  ralm  3325
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