MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reximssdv Structured version   Visualization version   GIF version

Theorem reximssdv 3173
Description: Derivation of a restricted existential quantification over a subset (the second hypothesis implies 𝐴𝐵), deduction form. (Contributed by AV, 21-Aug-2022.)
Hypotheses
Ref Expression
reximssdv.1 (𝜑 → ∃𝑥𝐵 𝜓)
reximssdv.2 ((𝜑 ∧ (𝑥𝐵𝜓)) → 𝑥𝐴)
reximssdv.3 ((𝜑 ∧ (𝑥𝐵𝜓)) → 𝜒)
Assertion
Ref Expression
reximssdv (𝜑 → ∃𝑥𝐴 𝜒)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem reximssdv
StepHypRef Expression
1 reximssdv.1 . 2 (𝜑 → ∃𝑥𝐵 𝜓)
2 reximssdv.2 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝜓)) → 𝑥𝐴)
3 reximssdv.3 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝜓)) → 𝜒)
42, 3jca 511 . . . 4 ((𝜑 ∧ (𝑥𝐵𝜓)) → (𝑥𝐴𝜒))
54ex 412 . . 3 (𝜑 → ((𝑥𝐵𝜓) → (𝑥𝐴𝜒)))
65reximdv2 3164 . 2 (𝜑 → (∃𝑥𝐵 𝜓 → ∃𝑥𝐴 𝜒))
71, 6mpd 15 1 (𝜑 → ∃𝑥𝐴 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-rex 3071
This theorem is referenced by:  ttrcltr  9756  fin1a2lem6  10445  fpwwe2lem11  10681  pgpssslw  19632  efgrelexlemb  19768  lspsneq  21124  lbsextlem4  21163  neissex  23135  iscnp4  23271  nlly2i  23484  llynlly  23485  qtophmeo  23825  ovolicc2lem5  25556  itgsubst  26090  footexALT  28726  footex  28729  opphllem1  28755  irngnzply1  33741  weiunfr  36468  lcfl6  41502  mapdval2N  41632  mapdpglem2  41675  hdmaprnlem10N  41861  primrootsunit1  42098  aks6d1c2  42131  aks6d1c6lem5  42178  aks5lem8  42202  pellfundglb  42896  oawordex2  43339  upciclem4  48926
  Copyright terms: Public domain W3C validator