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Theorem 19.38 1655
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 1472 . . 3 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
2 hba1 1521 . . 3 (∀𝑥𝜓 → ∀𝑥𝑥𝜓)
31, 2hbim 1525 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
4 19.8a 1570 . . 3 (𝜑 → ∃𝑥𝜑)
5 ax-4 1488 . . 3 (∀𝑥𝜓𝜓)
64, 5imim12i 59 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → (𝜑𝜓))
73, 6alrimih 1446 1 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1330  wex 1469
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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-ial 1515  ax-i5r 1516
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  19.23t  1656  sbi2v  1865  mo3h  2053  rgenm  3470  ralm  3472
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