Theorem elinintab 40025

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 4157	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ ∩ {𝑥 ∣ 𝜑}))
2		elintabg 40024	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))
3	2	pm5.32i 577	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))
4	1, 3	bitri 277	1 ⊢ (𝐴 ∈ (𝐵 ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   ∈ wcel 2114  {cab 2799   ∩ cin 3923  ∩ cint 4862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3488  df-in 3931  df-int 4863

This theorem is referenced by:  inintabss  40028  inintabd  40029  elcnvcnvintab  40032