Theorem elinintab 43042

Description: Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 13-Aug-2020.)

Assertion

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

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem elinintab

Step	Hyp	Ref	Expression
1		elin 3955	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ ∩ {𝑥 ∣ 𝜑}))
2		elintabg 4953	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))
3	2	pm5.32i 573	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))
4	1, 3	bitri 274	1 ⊢ (𝐴 ∈ (𝐵 ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1531   ∈ wcel 2098  {cab 2702   ∩ cin 3938  ∩ cint 4942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-v 3465  df-in 3946  df-int 4943

This theorem is referenced by:  inintabss  43045  inintabd  43046  elcnvcnvintab  43049