Theorem elintabg 4917

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 4914	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ∈ 𝑦))
2		eleq2w 2844	. . 3 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥))
3	2	ralab2 3654	. 2 ⊢ (∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ∈ 𝑦 ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))
4	1, 3	bitrdi 290	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∩ cint 4906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-int 4907

This theorem is used by:  elintab  4918  intidg  5424  elinintab  44576