Theorem elinintab 43931

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 3919	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ ∩ {𝑥 ∣ 𝜑}))
2		elintabg 4915	. . 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 1540   ∈ wcel 2114  {cab 2715   ∩ cin 3902  ∩ cint 4904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-v 3444  df-in 3910  df-int 4905

This theorem is referenced by:  inintabss  43934  inintabd  43935  elcnvcnvintab  43938