Theorem inintabd 44116

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		pm5.5 363	. . . . . 6 ⊢ (∃𝑥𝜓 → ((∃𝑥𝜓 → 𝑢 ∈ 𝐴) ↔ 𝑢 ∈ 𝐴))
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → ((∃𝑥𝜓 → 𝑢 ∈ 𝐴) ↔ 𝑢 ∈ 𝐴))
4	3	bicomd 225	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝐴 ↔ (∃𝑥𝜓 → 𝑢 ∈ 𝐴)))
5	4	anbi1d 640	. . 3 ⊢ (𝜑 → ((𝑢 ∈ 𝐴 ∧ ∀𝑥(𝜓 → 𝑢 ∈ 𝑥)) ↔ ((∃𝑥𝜓 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜓 → 𝑢 ∈ 𝑥))))
6		elinintab 44112	. . 3 ⊢ (𝑢 ∈ (𝐴 ∩ ∩ {𝑥 ∣ 𝜓}) ↔ (𝑢 ∈ 𝐴 ∧ ∀𝑥(𝜓 → 𝑢 ∈ 𝑥)))
7		elinintrab 44114	. . . 4 ⊢ (𝑢 ∈ V → (𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜓)} ↔ ((∃𝑥𝜓 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜓 → 𝑢 ∈ 𝑥))))
8	7	elv 3458	. . 3 ⊢ (𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜓)} ↔ ((∃𝑥𝜓 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜓 → 𝑢 ∈ 𝑥)))
9	5, 6, 8	3bitr4g 316	. 2 ⊢ (𝜑 → (𝑢 ∈ (𝐴 ∩ ∩ {𝑥 ∣ 𝜓}) ↔ 𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜓)}))
10	9	eqrdv 2759	1 ⊢ (𝜑 → (𝐴 ∩ ∩ {𝑥 ∣ 𝜓}) = ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜓)})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739  {crab 3413  Vcvv 3453   ∩ cin 3901  𝒫 cpw 4552  ∩ cint 4902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-in 3909  df-ss 3919  df-pw 4554  df-int 4903

This theorem is referenced by:  xpinintabd  44117