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Theorem inintabd 43569
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 pm5.5 361 . . . . . 6 (∃𝑥𝜓 → ((∃𝑥𝜓𝑢𝐴) ↔ 𝑢𝐴))
31, 2syl 17 . . . . 5 (𝜑 → ((∃𝑥𝜓𝑢𝐴) ↔ 𝑢𝐴))
43bicomd 223 . . . 4 (𝜑 → (𝑢𝐴 ↔ (∃𝑥𝜓𝑢𝐴)))
54anbi1d 631 . . 3 (𝜑 → ((𝑢𝐴 ∧ ∀𝑥(𝜓𝑢𝑥)) ↔ ((∃𝑥𝜓𝑢𝐴) ∧ ∀𝑥(𝜓𝑢𝑥))))
6 elinintab 43565 . . 3 (𝑢 ∈ (𝐴 {𝑥𝜓}) ↔ (𝑢𝐴 ∧ ∀𝑥(𝜓𝑢𝑥)))
7 elinintrab 43567 . . . 4 (𝑢 ∈ V → (𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜓)} ↔ ((∃𝑥𝜓𝑢𝐴) ∧ ∀𝑥(𝜓𝑢𝑥))))
87elv 3483 . . 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 2733 1 (𝜑 → (𝐴 {𝑥𝜓}) = {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜓)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106  {cab 2712  {crab 3433  Vcvv 3478  cin 3962  𝒫 cpw 4605   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-pw 4607  df-int 4952
This theorem is referenced by:  xpinintabd  43570
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