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Theorem bj-inrab 33229
Description: Generalization of inrab 4042. (Contributed by BJ, 21-Apr-2019.)
Assertion
Ref Expression
bj-inrab ({𝑥𝐴𝜑} ∩ {𝑥𝐵𝜓}) = {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}

Proof of Theorem bj-inrab
StepHypRef Expression
1 an4 900 . . . 4 (((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝜑𝜓)))
2 elin 3939 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
32anbi1i 733 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ (𝜑𝜓)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝜑𝜓)))
41, 3bitr4i 267 . . 3 (((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ (𝜑𝜓)))
54abbii 2877 . 2 {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓))} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ (𝜑𝜓))}
6 df-rab 3059 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
7 df-rab 3059 . . . 4 {𝑥𝐵𝜓} = {𝑥 ∣ (𝑥𝐵𝜓)}
86, 7ineq12i 3955 . . 3 ({𝑥𝐴𝜑} ∩ {𝑥𝐵𝜓}) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐵𝜓)})
9 inab 4038 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐵𝜓)}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓))}
108, 9eqtri 2782 . 2 ({𝑥𝐴𝜑} ∩ {𝑥𝐵𝜓}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓))}
11 df-rab 3059 . 2 {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ (𝜑𝜓))}
125, 10, 113eqtr4i 2792 1 ({𝑥𝐴𝜑} ∩ {𝑥𝐵𝜓}) = {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wcel 2139  {cab 2746  {crab 3054  cin 3714
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-rab 3059  df-v 3342  df-in 3722
This theorem is referenced by:  bj-inrab2  33230
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