ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssrexv GIF version

Theorem ssrexv 3075
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 3008 . . 3 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21anim1d 329 . 2 (𝐴𝐵 → ((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
32reximdv2 2468 1 (𝐴𝐵 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1436  wrex 2356  wss 2988
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-rex 2361  df-in 2994  df-ss 3001
This theorem is referenced by:  iunss1  3726  moriotass  5599  tfr1onlemssrecs  6060  tfrcllemssrecs  6073  supelti  6644  lbzbi  9036  rexico  10553  alzdvds  10761  zsupcl  10849  infssuzex  10851  gcddvds  10861  dvdslegcd  10862  bj-nn0suc  11328
  Copyright terms: Public domain W3C validator