Theorem inintabd 43541

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 ∩ ∩ {𝑥 ∣ 𝜓}) = ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 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.

Step	Hyp	Ref	Expression
1		inintabd.x	. . . . . 6 ⊢ (𝜑 → ∃𝑥𝜓)
2		pm5.5 361	. . . . . 6 ⊢ (∃𝑥𝜓 → ((∃𝑥𝜓 → 𝑢 ∈ 𝐴) ↔ 𝑢 ∈ 𝐴))
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → ((∃𝑥𝜓 → 𝑢 ∈ 𝐴) ↔ 𝑢 ∈ 𝐴))
4	3	bicomd 223	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝐴 ↔ (∃𝑥𝜓 → 𝑢 ∈ 𝐴)))
5	4	anbi1d 630	. . 3 ⊢ (𝜑 → ((𝑢 ∈ 𝐴 ∧ ∀𝑥(𝜓 → 𝑢 ∈ 𝑥)) ↔ ((∃𝑥𝜓 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜓 → 𝑢 ∈ 𝑥))))
6		elinintab 43537	. . 3 ⊢ (𝑢 ∈ (𝐴 ∩ ∩ {𝑥 ∣ 𝜓}) ↔ (𝑢 ∈ 𝐴 ∧ ∀𝑥(𝜓 → 𝑢 ∈ 𝑥)))
7		elinintrab 43539	. . . 4 ⊢ (𝑢 ∈ V → (𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜓)} ↔ ((∃𝑥𝜓 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜓 → 𝑢 ∈ 𝑥))))
8	7	elv 3493	. . 3 ⊢ (𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜓)} ↔ ((∃𝑥𝜓 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜓 → 𝑢 ∈ 𝑥)))
9	5, 6, 8	3bitr4g 314	. 2 ⊢ (𝜑 → (𝑢 ∈ (𝐴 ∩ ∩ {𝑥 ∣ 𝜓}) ↔ 𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜓)}))
10	9	eqrdv 2738	1 ⊢ (𝜑 → (𝐴 ∩ ∩ {𝑥 ∣ 𝜓}) = ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜓)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717  {crab 3443  Vcvv 3488   ∩ cin 3975  𝒫 cpw 4622  ∩ cint 4970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-in 3983  df-ss 3993  df-pw 4624  df-int 4971

This theorem is referenced by:  xpinintabd  43542