Theorem 3sstr4g 3312

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
3sstr4g.1	⊢ (φ → A ⊆ B)
3sstr4g.2	⊢ C = A
3sstr4g.3	⊢ D = B

Assertion

Ref	Expression
3sstr4g	⊢ (φ → C ⊆ D)

Proof of Theorem 3sstr4g

Step	Hyp	Ref	Expression
1		3sstr4g.1	. 2 ⊢ (φ → A ⊆ B)
2		3sstr4g.2	. . 3 ⊢ C = A
3		3sstr4g.3	. . 3 ⊢ D = B
4	2, 3	sseq12i 3297	. 2 ⊢ (C ⊆ D ↔ A ⊆ B)
5	1, 4	sylibr 203	1 ⊢ (φ → C ⊆ D)

Syntax hints:   → wi 4   = wceq 1642   ⊆ wss 3257

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by:  rabss2  3349  unss2  3434  sslin  3481  pw1ss  4169  ssopab2  4712  xpss12  4855  coss1  4872  coss2  4873  cnvss  4885  rnss  4959  ssres  4990  ssres2  4991  imass1  5023  imass2  5024  mapss  6027  sbthlem1  6203