Theorem 3sstr3d 3268

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Hypotheses

Ref	Expression
3sstr3d.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3sstr3d.2	⊢ (𝜑 → 𝐴 = 𝐶)
3sstr3d.3	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
3sstr3d	⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Proof of Theorem 3sstr3d

Step	Hyp	Ref	Expression
1		3sstr3d.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		3sstr3d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐶)
3		3sstr3d.3	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
4	2, 3	sseq12d 3255	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷))
5	1, 4	mpbid 147	1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Syntax hints:   → wi 4   = wceq 1395   ⊆ wss 3197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210

This theorem is referenced by:  fnsnsplitss  5837