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Theorem ssrexv 3162
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 3091 . . 3 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21anim1d 334 . 2 (𝐴𝐵 → ((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
32reximdv2 2531 1 (𝐴𝐵 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1480  wrex 2417  wss 3071
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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-rex 2422  df-in 3077  df-ss 3084
This theorem is referenced by:  iunss1  3824  moriotass  5758  tfr1onlemssrecs  6236  tfrcllemssrecs  6249  fiss  6865  supelti  6889  ctssdclemn0  6995  ctssdc  6998  enumctlemm  6999  lbzbi  9422  rexico  11007  alzdvds  11565  zsupcl  11653  infssuzex  11655  gcddvds  11665  dvdslegcd  11666  ssrest  12367  reeff1olem  12877  bj-nn0suc  13269
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