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Theorem 19.23t 1583
Description: Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
Assertion
Ref Expression
19.23t (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))

Proof of Theorem 19.23t
StepHypRef Expression
1 exim 1506 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
2 19.9t 1549 . . . 4 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
32biimpd 136 . . 3 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
41, 3syl9r 71 . 2 (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) → (∃𝑥𝜑𝜓)))
5 nfr 1427 . . . 4 (Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
65imim2d 52 . . 3 (Ⅎ𝑥𝜓 → ((∃𝑥𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
7 19.38 1582 . . 3 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
86, 7syl6 33 . 2 (Ⅎ𝑥𝜓 → ((∃𝑥𝜑𝜓) → ∀𝑥(𝜑𝜓)))
94, 8impbid 124 1 (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257  wnf 1365  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  19.23  1584  r19.23t  2440  ceqsalt  2597  vtoclgft  2621  sbciegft  2816
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