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

Theorem ssrexv 3005
 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 2939 . . 3 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21anim1d 319 . 2 (𝐴𝐵 → ((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
32reximdv2 2418 1 (𝐴𝐵 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393  ∃wrex 2307   ⊆ wss 2917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-rex 2312  df-in 2924  df-ss 2931 This theorem is referenced by:  iunss1  3668  moriotass  5496  lbzbi  8551  bj-nn0suc  10089
 Copyright terms: Public domain W3C validator