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Theorem 19.29r 1621
Description: Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
Assertion
Ref Expression
19.29r ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))

Proof of Theorem 19.29r
StepHypRef Expression
1 19.29 1620 . 2 ((∀𝑥𝜓 ∧ ∃𝑥𝜑) → ∃𝑥(𝜓𝜑))
2 ancom 266 . 2 ((∃𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (∀𝑥𝜓 ∧ ∃𝑥𝜑))
3 exancom 1608 . 2 (∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))
41, 2, 33imtr4i 201 1 ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1351  wex 1492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  19.29r2  1622  19.29x  1623  exan  1693  ax9o  1698  equvini  1758  eu2  2070  intab  3873  imadiflem  5295  bj-inex  14579
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