Theorem syl5eqssr 3317

Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
syl5eqssr.1	⊢ B = A
syl5eqssr.2	⊢ (φ → B ⊆ C)

Assertion

Ref	Expression
syl5eqssr	⊢ (φ → A ⊆ C)

Proof of Theorem syl5eqssr

Step	Hyp	Ref	Expression
1		syl5eqssr.1	. . 3 ⊢ B = A
2	1	eqcomi 2357	. 2 ⊢ A = B
3		syl5eqssr.2	. 2 ⊢ (φ → B ⊆ C)
4	2, 3	syl5eqss 3316	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:  cnvtr  5099  fimacnvdisj  5245  fimacnv  5412  dmen  6042  enprmaplem6  6082