Theorem ineqcomi 35520

Description: Disjointness inference (when 𝐶 = ∅), inference form of ineqcom 35519. (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 35519	. 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐶 ↔ (𝐵 ∩ 𝐴) = 𝐶)
3	1, 2	mpbi 232	1 ⊢ (𝐵 ∩ 𝐴) = 𝐶

Syntax hints:   = wceq 1537   ∩ cin 3935

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-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1540  df-ex 1781  df-sb 2070  df-clab 2800  df-cleq 2814  df-rab 3147  df-in 3943

This theorem is referenced by:  inres2  35521