Theorem syl6sseqr 3319

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

Hypotheses

Ref	Expression
syl6ssr.1	⊢ (φ → A ⊆ B)
syl6ssr.2	⊢ C = B

Assertion

Ref	Expression
syl6sseqr	⊢ (φ → A ⊆ C)

Proof of Theorem syl6sseqr

Step	Hyp	Ref	Expression
1		syl6ssr.1	. 2 ⊢ (φ → A ⊆ B)
2		syl6ssr.2	. . 3 ⊢ C = B
3	2	eqcomi 2357	. 2 ⊢ B = C
4	1, 3	syl6sseq 3318	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:  ncssfin  6152  sbthlem1  6204