Theorem ineqcomi 4174

Description: Two ways of expressing that two classes have a given intersection. Inference form of ineqcom 4173. Disjointness inference when 𝐶 = ∅. (Contributed by Peter Mazsa, 26-Mar-2017.) (Proof shortened by SN, 20-Sep-2024.)

Hypothesis

Ref	Expression
ineqcomi.1	⊢ (𝐴 ∩ 𝐵) = 𝐶

Assertion

Ref	Expression
ineqcomi	⊢ (𝐵 ∩ 𝐴) = 𝐶

Proof of Theorem ineqcomi

Step	Hyp	Ref	Expression
1		incom 4172	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
2		ineqcomi.1	. 2 ⊢ (𝐴 ∩ 𝐵) = 𝐶
3	1, 2	eqtri 2752	1 ⊢ (𝐵 ∩ 𝐴) = 𝐶

Syntax hints:   = wceq 1540   ∩ cin 3913

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 2008  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-rab 3406  df-in 3921

This theorem is referenced by:  cnvimainrn  7039  cnfldfunALT  21279  cnfldfunALTOLD  21292  psdmul  22053  vonf1owev  35095  inres2  38234  readvrec  42350  isubgr0uhgr  47873