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Theorem ssrexv 3060
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 2994 . . 3 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21anim1d 329 . 2 (𝐴𝐵 → ((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
32reximdv2 2461 1 (𝐴𝐵 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  wrex 2350  wss 2974
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-rex 2355  df-in 2980  df-ss 2987
This theorem is referenced by:  iunss1  3697  moriotass  5527  tfr1onlemssrecs  5988  tfrcllemssrecs  6001  supelti  6474  lbzbi  8782  rexico  10245  alzdvds  10399  zsupcl  10487  infssuzex  10489  gcddvds  10499  dvdslegcd  10500  bj-nn0suc  10917
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