Theorem inintabd 43012

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		pm5.5 360	. . . . . 6 ⊢ (∃𝑥𝜓 → ((∃𝑥𝜓 → 𝑢 ∈ 𝐴) ↔ 𝑢 ∈ 𝐴))
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → ((∃𝑥𝜓 → 𝑢 ∈ 𝐴) ↔ 𝑢 ∈ 𝐴))
4	3	bicomd 222	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝐴 ↔ (∃𝑥𝜓 → 𝑢 ∈ 𝐴)))
5	4	anbi1d 629	. . 3 ⊢ (𝜑 → ((𝑢 ∈ 𝐴 ∧ ∀𝑥(𝜓 → 𝑢 ∈ 𝑥)) ↔ ((∃𝑥𝜓 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜓 → 𝑢 ∈ 𝑥))))
6		elinintab 43008	. . 3 ⊢ (𝑢 ∈ (𝐴 ∩ ∩ {𝑥 ∣ 𝜓}) ↔ (𝑢 ∈ 𝐴 ∧ ∀𝑥(𝜓 → 𝑢 ∈ 𝑥)))
7		elinintrab 43010	. . . 4 ⊢ (𝑢 ∈ V → (𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜓)} ↔ ((∃𝑥𝜓 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜓 → 𝑢 ∈ 𝑥))))
8	7	elv 3477	. . 3 ⊢ (𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜓)} ↔ ((∃𝑥𝜓 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜓 → 𝑢 ∈ 𝑥)))
9	5, 6, 8	3bitr4g 313	. 2 ⊢ (𝜑 → (𝑢 ∈ (𝐴 ∩ ∩ {𝑥 ∣ 𝜓}) ↔ 𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜓)}))
10	9	eqrdv 2725	1 ⊢ (𝜑 → (𝐴 ∩ ∩ {𝑥 ∣ 𝜓}) = ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜓)})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1531   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2704  {crab 3428  Vcvv 3471   ∩ cin 3946  𝒫 cpw 4604  ∩ cint 4951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-in 3954  df-ss 3964  df-pw 4606  df-int 4952

This theorem is referenced by:  xpinintabd  43013