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Theorem inintabd 41858
Description: Value of the intersection of class with the intersection of a nonempty class. (Contributed by RP, 13-Aug-2020.)
Hypothesis
Ref Expression
inintabd.x (𝜑 → ∃𝑥𝜓)
Assertion
Ref Expression
inintabd (𝜑 → (𝐴 {𝑥𝜓}) = {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜓)})
Distinct variable groups:   𝜓,𝑤   𝑥,𝑤,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑤)   𝜓(𝑥)

Proof of Theorem inintabd
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 inintabd.x . . . . . 6 (𝜑 → ∃𝑥𝜓)
2 pm5.5 362 . . . . . 6 (∃𝑥𝜓 → ((∃𝑥𝜓𝑢𝐴) ↔ 𝑢𝐴))
31, 2syl 17 . . . . 5 (𝜑 → ((∃𝑥𝜓𝑢𝐴) ↔ 𝑢𝐴))
43bicomd 222 . . . 4 (𝜑 → (𝑢𝐴 ↔ (∃𝑥𝜓𝑢𝐴)))
54anbi1d 631 . . 3 (𝜑 → ((𝑢𝐴 ∧ ∀𝑥(𝜓𝑢𝑥)) ↔ ((∃𝑥𝜓𝑢𝐴) ∧ ∀𝑥(𝜓𝑢𝑥))))
6 elinintab 41854 . . 3 (𝑢 ∈ (𝐴 {𝑥𝜓}) ↔ (𝑢𝐴 ∧ ∀𝑥(𝜓𝑢𝑥)))
7 elinintrab 41856 . . . 4 (𝑢 ∈ V → (𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜓)} ↔ ((∃𝑥𝜓𝑢𝐴) ∧ ∀𝑥(𝜓𝑢𝑥))))
87elv 3452 . . 3 (𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜓)} ↔ ((∃𝑥𝜓𝑢𝐴) ∧ ∀𝑥(𝜓𝑢𝑥)))
95, 6, 83bitr4g 314 . 2 (𝜑 → (𝑢 ∈ (𝐴 {𝑥𝜓}) ↔ 𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜓)}))
109eqrdv 2735 1 (𝜑 → (𝐴 {𝑥𝜓}) = {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜓)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  {cab 2714  {crab 3408  Vcvv 3446  cin 3910  𝒫 cpw 4561   cint 4908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3409  df-v 3448  df-in 3918  df-ss 3928  df-pw 4563  df-int 4909
This theorem is referenced by:  xpinintabd  41859
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