Theorem elintrabg 4926

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 2852	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑}))
2		eleq1 2852	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
3	2	imbi2d 343	. . 3 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝑦 ∈ 𝑥) ↔ (𝜑 → 𝐴 ∈ 𝑥)))
4	3	ralbidv 3189	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐵 (𝜑 → 𝑦 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥)))
5		vex 3458	. . 3 ⊢ 𝑦 ∈ V
6	5	elintrab 4925	. 2 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝑦 ∈ 𝑥))
7	1, 4, 6	vtoclbg 3523	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  ∩ cint 4912

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-int 4913

This theorem is used by:  tskmid  10964  eltskm  10967  ldsysgenld  34904  ldgenpisyslem1  34907  elpcliN  41216  elmapintrab  44838