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Theorem rexss 3214
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 3141 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21pm4.71rd 392 . . . 4 (𝐴𝐵 → (𝑥𝐴 ↔ (𝑥𝐵𝑥𝐴)))
32anbi1d 462 . . 3 (𝐴𝐵 → ((𝑥𝐴𝜑) ↔ ((𝑥𝐵𝑥𝐴) ∧ 𝜑)))
4 anass 399 . . 3 (((𝑥𝐵𝑥𝐴) ∧ 𝜑) ↔ (𝑥𝐵 ∧ (𝑥𝐴𝜑)))
53, 4bitrdi 195 . 2 (𝐴𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵 ∧ (𝑥𝐴𝜑))))
65rexbidv2 2473 1 (𝐴𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 (𝑥𝐴𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 2141  wrex 2449  wss 3121
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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-rex 2454  df-in 3127  df-ss 3134
This theorem is referenced by:  1idprl  7552  1idpru  7553  ltexprlemm  7562  suplocexprlemmu  7680  oddnn02np1  11839  oddge22np1  11840  evennn02n  11841  evennn2n  11842
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