Metamath Proof Explorer
Description: Substitution of equality into both sides of a subclass relationship.
(Contributed by NM, 1-Oct-2000)
|
|
Ref |
Expression |
|
Hypotheses |
3sstr3d.1 |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 ) |
|
|
3sstr3d.2 |
⊢ ( 𝜑 → 𝐴 = 𝐶 ) |
|
|
3sstr3d.3 |
⊢ ( 𝜑 → 𝐵 = 𝐷 ) |
|
Assertion |
3sstr3d |
⊢ ( 𝜑 → 𝐶 ⊆ 𝐷 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
3sstr3d.1 |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 ) |
2 |
|
3sstr3d.2 |
⊢ ( 𝜑 → 𝐴 = 𝐶 ) |
3 |
|
3sstr3d.3 |
⊢ ( 𝜑 → 𝐵 = 𝐷 ) |
4 |
2 3
|
sseq12d |
⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷 ) ) |
5 |
1 4
|
mpbid |
⊢ ( 𝜑 → 𝐶 ⊆ 𝐷 ) |