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Theorem r19.2m 3353
Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1572). 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 2360 . . . 4 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
2 exintr 1568 . . . 4 (∀𝑥(𝑥𝐴𝜑) → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝜑)))
31, 2sylbi 119 . . 3 (∀𝑥𝐴 𝜑 → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝜑)))
4 df-rex 2361 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
53, 4syl6ibr 160 . 2 (∀𝑥𝐴 𝜑 → (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 𝜑))
65impcom 123 1 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1285  wex 1424  wcel 1436  wral 2355  wrex 2356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-ral 2360  df-rex 2361
This theorem is referenced by:  intssunim  3687  riinm  3779  trintssmOLD  3921  iinexgm  3958  xpiindim  4534  cnviinm  4929  eusvobj2  5580  iinerm  6297  rexfiuz  10263  r19.2uz  10267  climuni  10520
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