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Theorem rexss 3810
 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 3738 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21pm4.71rd 670 . . . 4 (𝐴𝐵 → (𝑥𝐴 ↔ (𝑥𝐵𝑥𝐴)))
32anbi1d 743 . . 3 (𝐴𝐵 → ((𝑥𝐴𝜑) ↔ ((𝑥𝐵𝑥𝐴) ∧ 𝜑)))
4 anass 684 . . 3 (((𝑥𝐵𝑥𝐴) ∧ 𝜑) ↔ (𝑥𝐵 ∧ (𝑥𝐴𝜑)))
53, 4syl6bb 276 . 2 (𝐴𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵 ∧ (𝑥𝐴𝜑))))
65rexbidv2 3186 1 (𝐴𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 (𝑥𝐴𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∈ wcel 2139  ∃wrex 3051   ⊆ wss 3715 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-rex 3056  df-in 3722  df-ss 3729 This theorem is referenced by:  oddnn02np1  15294  oddge22np1  15295  evennn02n  15296  evennn2n  15297  2lgslem1a  25336  omssubadd  30692  limsupmnfuzlem  40479  sbgoldbo  42203
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