Theorem elinintab 43599

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 3942	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ ∩ {𝑥 ∣ 𝜑}))
2		elintabg 4933	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))
3	2	pm5.32i 574	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))
4	1, 3	bitri 275	1 ⊢ (𝐴 ∈ (𝐵 ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   ∈ wcel 2108  {cab 2713   ∩ cin 3925  ∩ cint 4922

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 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-v 3461  df-in 3933  df-int 4923

This theorem is referenced by:  inintabss  43602  inintabd  43603  elcnvcnvintab  43606