Theorem ssinss2d 46185

Description: Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
ssinss2d.1	⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Assertion

Ref	Expression
ssinss2d	⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)

Proof of Theorem ssinss2d

Step	Hyp	Ref	Expression
1		incom 4105	. 2 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
2		ssinss2d.1	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
3	2	ssinss1d 4143	. 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ⊆ 𝐶)
4	1, 3	eqsstrid 3947	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3896   ∩ cin 4088

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-in 4090

This theorem is used by:  fourierdlem49  47273  caragenuncllem  47630