Theorem ssind 3155

Description: A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
ssind.1	⊢ (φ → A ⊆ B)
ssind.2	⊢ (φ → A ⊆ 𝐶)

Assertion

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

Proof of Theorem ssind

Step	Hyp	Ref	Expression
1		ssind.1	. 2 ⊢ (φ → A ⊆ B)
2		ssind.2	. 2 ⊢ (φ → A ⊆ 𝐶)
3		ssin 3153	. . 3 ⊢ ((A ⊆ B ∧ A ⊆ 𝐶) ↔ A ⊆ (B ∩ 𝐶))
4	3	biimpi 113	. 2 ⊢ ((A ⊆ B ∧ A ⊆ 𝐶) → A ⊆ (B ∩ 𝐶))
5	1, 2, 4	syl2anc 391	1 ⊢ (φ → A ⊆ (B ∩ 𝐶))

Syntax hints:   → wi 4   ∧ wa 97   ∩ cin 2910   ⊆ wss 2911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925

This theorem is referenced by: (None)