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Theorem bj-inrab 34676
 Description: Generalization of inrab 4211. (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 655 . . . 4 (((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝜑𝜓)))
2 elin 3876 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
32anbi1i 626 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ (𝜑𝜓)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝜑𝜓)))
41, 3bitr4i 281 . . 3 (((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ (𝜑𝜓)))
54abbii 2823 . 2 {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓))} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ (𝜑𝜓))}
6 df-rab 3079 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
7 df-rab 3079 . . . 4 {𝑥𝐵𝜓} = {𝑥 ∣ (𝑥𝐵𝜓)}
86, 7ineq12i 4117 . . 3 ({𝑥𝐴𝜑} ∩ {𝑥𝐵𝜓}) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐵𝜓)})
9 inab 4205 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐵𝜓)}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓))}
108, 9eqtri 2781 . 2 ({𝑥𝐴𝜑} ∩ {𝑥𝐵𝜓}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓))}
11 df-rab 3079 . 2 {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ (𝜑𝜓))}
125, 10, 113eqtr4i 2791 1 ({𝑥𝐴𝜑} ∩ {𝑥𝐵𝜓}) = {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  {crab 3074   ∩ cin 3859 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 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-rab 3079  df-v 3411  df-in 3867 This theorem is referenced by:  bj-inrab2  34677
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