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Theorem rexss 3195
Description: Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
rexss (𝐴𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 (𝑥𝐴𝜑)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexss
StepHypRef Expression
1 ssel 3122 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21pm4.71rd 392 . . . 4 (𝐴𝐵 → (𝑥𝐴 ↔ (𝑥𝐵𝑥𝐴)))
32anbi1d 461 . . 3 (𝐴𝐵 → ((𝑥𝐴𝜑) ↔ ((𝑥𝐵𝑥𝐴) ∧ 𝜑)))
4 anass 399 . . 3 (((𝑥𝐵𝑥𝐴) ∧ 𝜑) ↔ (𝑥𝐵 ∧ (𝑥𝐴𝜑)))
53, 4bitrdi 195 . 2 (𝐴𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵 ∧ (𝑥𝐴𝜑))))
65rexbidv2 2460 1 (𝐴𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 (𝑥𝐴𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 2128  wrex 2436  wss 3102
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-rex 2441  df-in 3108  df-ss 3115
This theorem is referenced by:  1idprl  7510  1idpru  7511  ltexprlemm  7520  suplocexprlemmu  7638  oddnn02np1  11771  oddge22np1  11772  evennn02n  11773  evennn2n  11774
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