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Theorem inintabd 40266
 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 pm5.5 365 . . . . . 6 (∃𝑥𝜓 → ((∃𝑥𝜓𝑢𝐴) ↔ 𝑢𝐴))
31, 2syl 17 . . . . 5 (𝜑 → ((∃𝑥𝜓𝑢𝐴) ↔ 𝑢𝐴))
43bicomd 226 . . . 4 (𝜑 → (𝑢𝐴 ↔ (∃𝑥𝜓𝑢𝐴)))
54anbi1d 632 . . 3 (𝜑 → ((𝑢𝐴 ∧ ∀𝑥(𝜓𝑢𝑥)) ↔ ((∃𝑥𝜓𝑢𝐴) ∧ ∀𝑥(𝜓𝑢𝑥))))
6 elinintab 40262 . . 3 (𝑢 ∈ (𝐴 {𝑥𝜓}) ↔ (𝑢𝐴 ∧ ∀𝑥(𝜓𝑢𝑥)))
7 elinintrab 40264 . . . 4 (𝑢 ∈ V → (𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜓)} ↔ ((∃𝑥𝜓𝑢𝐴) ∧ ∀𝑥(𝜓𝑢𝑥))))
87elv 3449 . . 3 (𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜓)} ↔ ((∃𝑥𝜓𝑢𝐴) ∧ ∀𝑥(𝜓𝑢𝑥)))
95, 6, 83bitr4g 317 . 2 (𝜑 → (𝑢 ∈ (𝐴 {𝑥𝜓}) ↔ 𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜓)}))
109eqrdv 2799 1 (𝜑 → (𝐴 {𝑥𝜓}) = {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜓)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2112  {cab 2779  {crab 3113  Vcvv 3444   ∩ cin 3883  𝒫 cpw 4500  ∩ cint 4841 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-rab 3118  df-v 3446  df-in 3891  df-ss 3901  df-pw 4502  df-int 4842 This theorem is referenced by:  xpinintabd  40267
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