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Theorem elinintab 38401
Description: Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 13-Aug-2020.)
Assertion
Ref Expression
elinintab (𝐴 ∈ (𝐵 {𝑥𝜑}) ↔ (𝐴𝐵 ∧ ∀𝑥(𝜑𝐴𝑥)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem elinintab
StepHypRef Expression
1 elin 3939 . 2 (𝐴 ∈ (𝐵 {𝑥𝜑}) ↔ (𝐴𝐵𝐴 {𝑥𝜑}))
2 elintabg 38400 . . 3 (𝐴𝐵 → (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥)))
32pm5.32i 672 . 2 ((𝐴𝐵𝐴 {𝑥𝜑}) ↔ (𝐴𝐵 ∧ ∀𝑥(𝜑𝐴𝑥)))
41, 3bitri 264 1 (𝐴 ∈ (𝐵 {𝑥𝜑}) ↔ (𝐴𝐵 ∧ ∀𝑥(𝜑𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630  wcel 2139  {cab 2746  cin 3714   cint 4627
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-nfc 2891  df-ral 3055  df-v 3342  df-in 3722  df-int 4628
This theorem is referenced by:  inintabss  38404  inintabd  38405  elcnvcnvintab  38408
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