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Theorem bj-inrab 33283
Description: Generalization of inrab 4063. (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 646 . . . 4 (((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝜑𝜓)))
2 elin 3958 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
32anbi1i 617 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ (𝜑𝜓)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝜑𝜓)))
41, 3bitr4i 269 . . 3 (((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ (𝜑𝜓)))
54abbii 2882 . 2 {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓))} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ (𝜑𝜓))}
6 df-rab 3064 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
7 df-rab 3064 . . . 4 {𝑥𝐵𝜓} = {𝑥 ∣ (𝑥𝐵𝜓)}
86, 7ineq12i 3974 . . 3 ({𝑥𝐴𝜑} ∩ {𝑥𝐵𝜓}) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐵𝜓)})
9 inab 4059 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ∩ {𝑥 ∣ (𝑥𝐵𝜓)}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓))}
108, 9eqtri 2787 . 2 ({𝑥𝐴𝜑} ∩ {𝑥𝐵𝜓}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ (𝑥𝐵𝜓))}
11 df-rab 3064 . 2 {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ (𝜑𝜓))}
125, 10, 113eqtr4i 2797 1 ({𝑥𝐴𝜑} ∩ {𝑥𝐵𝜓}) = {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1652  wcel 2155  {cab 2751  {crab 3059  cin 3731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rab 3064  df-v 3352  df-in 3739
This theorem is referenced by:  bj-inrab2  33284
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