Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.29r GIF version

Theorem 19.29r 1512
 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 1511 . 2 ((∀𝑥𝜓 ∧ ∃𝑥𝜑) → ∃𝑥(𝜓𝜑))
2 ancom 253 . 2 ((∃𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (∀𝑥𝜓 ∧ ∃𝑥𝜑))
3 exancom 1499 . 2 (∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))
41, 2, 33imtr4i 190 1 ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  19.29r2  1513  19.29x  1514  exan  1583  ax9o  1588  equvini  1641  eu2  1944  intab  3644  imadiflem  4978  bj-inex  10027
 Copyright terms: Public domain W3C validator