Theorem ineqcomi 3415

Description: Two ways of expressing that two classes have a given intersection. Inference form of ineqcom 3414. 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 3413	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
2		ineqcomi.1	. 2 ⊢ (𝐴 ∩ 𝐵) = 𝐶
3	1, 2	eqtri 2255	1 ⊢ (𝐵 ∩ 𝐴) = 𝐶

Syntax hints:   = wceq 1398   ∩ cin 3212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3219

This theorem is referenced by:  disjdifr  3584