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Theorem 19.38 1607
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 1425 . . 3 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
2 hba1 1474 . . 3 (∀𝑥𝜓 → ∀𝑥𝑥𝜓)
31, 2hbim 1478 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
4 19.8a 1523 . . 3 (𝜑 → ∃𝑥𝜑)
5 ax-4 1441 . . 3 (∀𝑥𝜓𝜓)
64, 5imim12i 58 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → (𝜑𝜓))
73, 6alrimih 1399 1 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  wex 1422
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  19.23t  1608  sbi2v  1814  mo3h  1995  rgenm  3351  ralm  3353
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