Theorem elintrabg 3884

Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)

Assertion

Ref	Expression
elintrabg	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥)))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem elintrabg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2256	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑}))
2		eleq1 2256	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
3	2	imbi2d 230	. . 3 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝑦 ∈ 𝑥) ↔ (𝜑 → 𝐴 ∈ 𝑥)))
4	3	ralbidv 2494	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐵 (𝜑 → 𝑦 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥)))
5		vex 2763	. . 3 ⊢ 𝑦 ∈ V
6	5	elintrab 3883	. 2 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝑦 ∈ 𝑥))
7	1, 4, 6	vtoclbg 2822	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ∀wral 2472  {crab 2476  ∩ cint 3871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rab 2481  df-v 2762  df-int 3872

This theorem is referenced by: (None)