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Theorem r19.2m 3449
Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1617). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) (Revised by Jim Kingdon, 7-Apr-2023.)
Assertion
Ref Expression
r19.2m ((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem r19.2m
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eleq1w 2200 . . . 4 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
21cbvexv 1890 . . 3 (∃𝑥 𝑥𝐴 ↔ ∃𝑧 𝑧𝐴)
3 eleq1w 2200 . . . 4 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
43cbvexv 1890 . . 3 (∃𝑧 𝑧𝐴 ↔ ∃𝑦 𝑦𝐴)
52, 4bitri 183 . 2 (∃𝑥 𝑥𝐴 ↔ ∃𝑦 𝑦𝐴)
6 df-ral 2421 . . . . 5 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
7 exintr 1613 . . . . 5 (∀𝑥(𝑥𝐴𝜑) → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝜑)))
86, 7sylbi 120 . . . 4 (∀𝑥𝐴 𝜑 → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝜑)))
9 df-rex 2422 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
108, 9syl6ibr 161 . . 3 (∀𝑥𝐴 𝜑 → (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 𝜑))
1110impcom 124 . 2 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)
125, 11sylanbr 283 1 ((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1329  wex 1468  wcel 1480  wral 2416  wrex 2417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-clel 2135  df-ral 2421  df-rex 2422
This theorem is referenced by:  intssunim  3793  riinm  3885  iinexgm  4079  xpiindim  4676  cnviinm  5080  eusvobj2  5760  iinerm  6501  suplocexprlemml  7531  rexfiuz  10768  r19.2uz  10772  climuni  11069  cncnp2m  12410
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