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Theorem 19.29r 1528
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 1527 . 2 ((∀𝑥𝜓 ∧ ∃𝑥𝜑) → ∃𝑥(𝜓𝜑))
2 ancom 257 . 2 ((∃𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (∀𝑥𝜓 ∧ ∃𝑥𝜑))
3 exancom 1515 . 2 (∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))
41, 2, 33imtr4i 194 1 ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257  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
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  19.29r2  1529  19.29x  1530  exan  1599  ax9o  1604  equvini  1657  eu2  1960  intab  3672  imadiflem  5006  bj-inex  10414
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