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Theorem r19.2m 3306
Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1529). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
Assertion
Ref Expression
r19.2m ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem r19.2m
StepHypRef Expression
1 df-ral 2308 . . . 4 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
2 exintr 1525 . . . 4 (∀𝑥(𝑥𝐴𝜑) → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝜑)))
31, 2sylbi 114 . . 3 (∀𝑥𝐴 𝜑 → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝜑)))
4 df-rex 2309 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
53, 4syl6ibr 151 . 2 (∀𝑥𝐴 𝜑 → (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 𝜑))
65impcom 116 1 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wal 1241  wex 1381  wcel 1393  wral 2303  wrex 2304
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  df-ral 2308  df-rex 2309
This theorem is referenced by:  intssunim  3634  riinm  3726  trintssm  3867  iinexgm  3905  xpiindim  4436  cnviinm  4822  eusvobj2  5461  iinerm  6141  r19.2uz  9445  climuni  9667
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