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Theorem elintabg 40204
 Description: Two ways of saying a set is an element of the intersection of a class. (Contributed by RP, 13-Aug-2020.)
Assertion
Ref Expression
elintabg (𝐴𝑉 → (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem elintabg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elintg 4859 . 2 (𝐴𝑉 → (𝐴 {𝑥𝜑} ↔ ∀𝑦 ∈ {𝑥𝜑}𝐴𝑦))
2 eleq2w 2897 . . 3 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
32ralab2 3663 . 2 (∀𝑦 ∈ {𝑥𝜑}𝐴𝑦 ↔ ∀𝑥(𝜑𝐴𝑥))
41, 3syl6bb 290 1 (𝐴𝑉 → (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2114  {cab 2800  ∀wral 3130  ∩ cint 4851 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-ral 3135  df-int 4852 This theorem is referenced by:  elinintab  40205
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