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Theorem reximssdv 2536
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 304 . . . 4 ((𝜑 ∧ (𝑥𝐵𝜓)) → (𝑥𝐴𝜒))
54ex 114 . . 3 (𝜑 → ((𝑥𝐵𝜓) → (𝑥𝐴𝜒)))
65reximdv2 2531 . 2 (𝜑 → (∃𝑥𝐵 𝜓 → ∃𝑥𝐴 𝜒))
71, 6mpd 13 1 (𝜑 → ∃𝑥𝐴 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1480  wrex 2417
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-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-rex 2422
This theorem is referenced by:  suplocexprlemrl  7537  neissex  12348  iscnp4  12401  suplociccex  12786
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