Theorem ssbrd 3805

Description: Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)

Hypothesis

Ref	Expression
ssbrd.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
ssbrd	⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))

Proof of Theorem ssbrd

Step	Hyp	Ref	Expression
1		ssbrd.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2	1	sseld 2944	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∈ 𝐴 → ⟨𝐶, 𝐷⟩ ∈ 𝐵))
3		df-br 3765	. 2 ⊢ (𝐶𝐴𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐴)
4		df-br 3765	. 2 ⊢ (𝐶𝐵𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐵)
5	2, 3, 4	3imtr4g 194	1 ⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))

Syntax hints:   → wi 4   ∈ wcel 1393   ⊆ wss 2917  ⟨cop 3378   class class class wbr 3764

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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-br 3765

This theorem is referenced by:  ssbri  3806  sess1  4074  brrelex12  4381  coss1  4491  coss2  4492  eqbrrdva  4505  ersym  6118  ertr  6121