Theorem ineqcomi 35930

Description: Disjointness inference (when 𝐶 = ∅), inference form of ineqcom 35929. (Contributed by Peter Mazsa, 26-Mar-2017.)

Hypothesis

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

Assertion

Ref	Expression
ineqcomi	⊢ (𝐵 ∩ 𝐴) = 𝐶

Proof of Theorem ineqcomi

Step	Hyp	Ref	Expression
1		ineqcomi.1	. 2 ⊢ (𝐴 ∩ 𝐵) = 𝐶
2		ineqcom 35929	. 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐶 ↔ (𝐵 ∩ 𝐴) = 𝐶)
3	1, 2	mpbi 233	1 ⊢ (𝐵 ∩ 𝐴) = 𝐶

Syntax hints:   = wceq 1539   ∩ cin 3853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2122  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-rab 3077  df-in 3861

This theorem is referenced by:  inres2  35931