Theorem sselda 2939

Description: Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)

Hypothesis

Ref	Expression
sseld.1	⊢ (φ → A ⊆ B)

Assertion

Ref	Expression
sselda	⊢ ((φ ∧ 𝐶 ∈ A) → 𝐶 ∈ B)

Proof of Theorem sselda

Step	Hyp	Ref	Expression
1		sseld.1	. . 3 ⊢ (φ → A ⊆ B)
2	1	sseld 2938	. 2 ⊢ (φ → (𝐶 ∈ A → 𝐶 ∈ B))
3	2	imp 115	1 ⊢ ((φ ∧ 𝐶 ∈ A) → 𝐶 ∈ B)

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390   ⊆ 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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  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-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925

This theorem is referenced by:  elrel  4385  ffvresb  5271  1stdm  5750  tfrlem1  5864  tfrlemiubacc  5885  erinxp  6116  fundmen  6222  elprnql  6464  elprnqu  6465  un0addcl  7991  un0mulcl  7992  icoshftf1o  8629  elfzom1elfzo  8829  zpnn0elfzo  8833  iseqfveq  8907