Theorem eqsstrd 3306

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
eqsstrd.1	⊢ (φ → A = B)
eqsstrd.2	⊢ (φ → B ⊆ C)

Assertion

Ref	Expression
eqsstrd	⊢ (φ → A ⊆ C)

Proof of Theorem eqsstrd

Step	Hyp	Ref	Expression
1		eqsstrd.2	. 2 ⊢ (φ → B ⊆ C)
2		eqsstrd.1	. . 3 ⊢ (φ → A = B)
3	2	sseq1d 3299	. 2 ⊢ (φ → (A ⊆ C ↔ B ⊆ C))
4	1, 3	mpbird 223	1 ⊢ (φ → A ⊆ C)

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:  eqsstr3d  3307  syl6eqss  3322  fimacnv  5412  fvmptss  5706  fvmptss2  5726  spacssnc  6285  spacind  6288