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Theorem bj-inrab 32597
Description: Generalization of inrab 3880. (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 864 . . . 4 (((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝜑𝜓)))
2 elin 3779 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
32anbi1i 730 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ (𝜑𝜓)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝜑𝜓)))
41, 3bitr4i 267 . . 3 (((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ (𝜑𝜓)))
54abbii 2736 . 2 {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓))} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ (𝜑𝜓))}
6 df-rab 2916 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
7 df-rab 2916 . . . 4 {𝑥𝐵𝜓} = {𝑥 ∣ (𝑥𝐵𝜓)}
86, 7ineq12i 3795 . . 3 ({𝑥𝐴𝜑} ∩ {𝑥𝐵𝜓}) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐵𝜓)})
9 inab 3876 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐵𝜓)}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓))}
108, 9eqtri 2643 . 2 ({𝑥𝐴𝜑} ∩ {𝑥𝐵𝜓}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓))}
11 df-rab 2916 . 2 {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ (𝜑𝜓))}
125, 10, 113eqtr4i 2653 1 ({𝑥𝐴𝜑} ∩ {𝑥𝐵𝜓}) = {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  {cab 2607  {crab 2911  cin 3558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3191  df-in 3566
This theorem is referenced by:  bj-inrab2  32598
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