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Theorem r19.2m 3500
Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1631). 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 2231 . . . 4 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
21cbvexv 1911 . . 3 (∃𝑥 𝑥𝐴 ↔ ∃𝑧 𝑧𝐴)
3 eleq1w 2231 . . . 4 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
43cbvexv 1911 . . 3 (∃𝑧 𝑧𝐴 ↔ ∃𝑦 𝑦𝐴)
52, 4bitri 183 . 2 (∃𝑥 𝑥𝐴 ↔ ∃𝑦 𝑦𝐴)
6 df-ral 2453 . . . . 5 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
7 exintr 1627 . . . . 5 (∀𝑥(𝑥𝐴𝜑) → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝜑)))
86, 7sylbi 120 . . . 4 (∀𝑥𝐴 𝜑 → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝜑)))
9 df-rex 2454 . . . 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 1346  wex 1485  wcel 2141  wral 2448  wrex 2449
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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-clel 2166  df-ral 2453  df-rex 2454
This theorem is referenced by:  intssunim  3851  riinm  3943  iinexgm  4138  xpiindim  4746  cnviinm  5150  eusvobj2  5836  iinerm  6581  suplocexprlemml  7665  rexfiuz  10940  r19.2uz  10944  climuni  11243  pc2dvds  12270  cncnp2m  12984
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