Theorem inex2g 5188

Description: Sufficient condition for an intersection to be a set. Commuted form of inex1g 5187. (Contributed by Peter Mazsa, 19-Dec-2018.)

Assertion

Ref	Expression
inex2g	⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V)

Proof of Theorem inex2g

Step	Hyp	Ref	Expression
1		incom 4128	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
2		inex1g 5187	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
3	1, 2	eqeltrid 2894	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2111  Vcvv 3441   ∩ cin 3880

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-sep 5167

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-in 3888

This theorem is referenced by:  satefvfmla1  32785  inex3  35755  inxpex  35756  dfcnvrefrels2  35926  dfcnvrefrels3  35927