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Theorem ssrexv 3109
Description: Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)
Assertion
Ref Expression
ssrexv (𝐴𝐵 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssrexv
StepHypRef Expression
1 ssel 3041 . . 3 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21anim1d 332 . 2 (𝐴𝐵 → ((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
32reximdv2 2490 1 (𝐴𝐵 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1448  wrex 2376  wss 3021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-rex 2381  df-in 3027  df-ss 3034
This theorem is referenced by:  iunss1  3771  moriotass  5690  tfr1onlemssrecs  6166  tfrcllemssrecs  6179  supelti  6804  ctssdclemn0  6910  ctssdc  6912  enumctlemm  6913  lbzbi  9258  rexico  10833  alzdvds  11347  zsupcl  11435  infssuzex  11437  gcddvds  11447  dvdslegcd  11448  ssrest  12133  bj-nn0suc  12747
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