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Theorem r19.2m 3509
Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1638). 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 2238 . . . 4 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
21cbvexv 1918 . . 3 (∃𝑥 𝑥𝐴 ↔ ∃𝑧 𝑧𝐴)
3 eleq1w 2238 . . . 4 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
43cbvexv 1918 . . 3 (∃𝑧 𝑧𝐴 ↔ ∃𝑦 𝑦𝐴)
52, 4bitri 184 . 2 (∃𝑥 𝑥𝐴 ↔ ∃𝑦 𝑦𝐴)
6 df-ral 2460 . . . . 5 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
7 exintr 1634 . . . . 5 (∀𝑥(𝑥𝐴𝜑) → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝜑)))
86, 7sylbi 121 . . . 4 (∀𝑥𝐴 𝜑 → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝜑)))
9 df-rex 2461 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
108, 9imbitrrdi 162 . . 3 (∀𝑥𝐴 𝜑 → (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 𝜑))
1110impcom 125 . 2 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)
125, 11sylanbr 285 1 ((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1351  wex 1492  wcel 2148  wral 2455  wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-clel 2173  df-ral 2460  df-rex 2461
This theorem is referenced by:  intssunim  3866  riinm  3959  iinexgm  4154  xpiindim  4764  cnviinm  5170  eusvobj2  5860  iinerm  6606  suplocexprlemml  7714  rexfiuz  10997  r19.2uz  11001  climuni  11300  pc2dvds  12328  issubg4m  13051  cncnp2m  13701
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