Theorem 3sstr4g 3313

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 3298	. 2 ⊢ (C ⊆ D ↔ A ⊆ B)
5	1, 4	sylibr 203	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:  rabss2  3350  unss2  3435  sslin  3482  pw1ss  4170  ssopab2  4713  xpss12  4856  coss1  4873  coss2  4874  cnvss  4886  rnss  4960  ssres  4991  ssres2  4992  imass1  5024  imass2  5025  mapss  6028  sbthlem1  6204