Theorem 3sstr4d 3315

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
3sstr4d.1	⊢ (φ → A ⊆ B)
3sstr4d.2	⊢ (φ → C = A)
3sstr4d.3	⊢ (φ → D = B)

Assertion

Ref	Expression
3sstr4d	⊢ (φ → C ⊆ D)

Proof of Theorem 3sstr4d

Step	Hyp	Ref	Expression
1		3sstr4d.1	. 2 ⊢ (φ → A ⊆ B)
2		3sstr4d.2	. . 3 ⊢ (φ → C = A)
3		3sstr4d.3	. . 3 ⊢ (φ → D = B)
4	2, 3	sseq12d 3301	. 2 ⊢ (φ → (C ⊆ D ↔ A ⊆ B))
5	1, 4	mpbird 223	1 ⊢ (φ → C ⊆ D)

Syntax hints:   → wi 4   = wceq 1642   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  mapsspm  6022  sbthlem3  6206