Theorem elintabg 4957

Description: Two ways of saying a set is an element of the intersection of a class. (Contributed by NM, 30-Aug-1993.) Put in closed form. (Revised by RP, 13-Aug-2020.)

Assertion

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

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem elintabg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elintg 4954	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ∈ 𝑦))
2		eleq2w 2825	. . 3 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥))
3	2	ralab2 3703	. 2 ⊢ (∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ∈ 𝑦 ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))
4	1, 3	bitrdi 287	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∩ cint 4946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-int 4947

This theorem is referenced by:  elintab  4958  intidg  5462  elinintab  43588