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Theorem inintabd 39279
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 pm5.5 354 . . . . . 6 (∃𝑥𝜓 → ((∃𝑥𝜓𝑢𝐴) ↔ 𝑢𝐴))
31, 2syl 17 . . . . 5 (𝜑 → ((∃𝑥𝜓𝑢𝐴) ↔ 𝑢𝐴))
43bicomd 215 . . . 4 (𝜑 → (𝑢𝐴 ↔ (∃𝑥𝜓𝑢𝐴)))
54anbi1d 620 . . 3 (𝜑 → ((𝑢𝐴 ∧ ∀𝑥(𝜓𝑢𝑥)) ↔ ((∃𝑥𝜓𝑢𝐴) ∧ ∀𝑥(𝜓𝑢𝑥))))
6 elinintab 39275 . . 3 (𝑢 ∈ (𝐴 {𝑥𝜓}) ↔ (𝑢𝐴 ∧ ∀𝑥(𝜓𝑢𝑥)))
7 elinintrab 39277 . . . 4 (𝑢 ∈ V → (𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜓)} ↔ ((∃𝑥𝜓𝑢𝐴) ∧ ∀𝑥(𝜓𝑢𝑥))))
87elv 3417 . . 3 (𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜓)} ↔ ((∃𝑥𝜓𝑢𝐴) ∧ ∀𝑥(𝜓𝑢𝑥)))
95, 6, 83bitr4g 306 . 2 (𝜑 → (𝑢 ∈ (𝐴 {𝑥𝜓}) ↔ 𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜓)}))
109eqrdv 2773 1 (𝜑 → (𝐴 {𝑥𝜓}) = {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜓)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wal 1505   = wceq 1507  wex 1742  wcel 2048  {cab 2755  {crab 3089  Vcvv 3412  cin 3827  𝒫 cpw 4420   cint 4747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2747  ax-sep 5058
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-ral 3090  df-rab 3094  df-v 3414  df-in 3835  df-ss 3842  df-pw 4422  df-int 4748
This theorem is referenced by:  xpinintabd  39280
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