Theorem syl6eqss 3321

Description: A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
syl6eqss.1	⊢ (φ → A = B)
syl6eqss.2	⊢ B ⊆ C

Assertion

Ref	Expression
syl6eqss	⊢ (φ → A ⊆ C)

Proof of Theorem syl6eqss

Step	Hyp	Ref	Expression
1		syl6eqss.1	. 2 ⊢ (φ → A = B)
2		syl6eqss.2	. . 3 ⊢ B ⊆ C
3	2	a1i 10	. 2 ⊢ (φ → B ⊆ C)
4	1, 3	eqsstrd 3305	1 ⊢ (φ → A ⊆ C)

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

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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by:  syl6eqssr  3322  dmsnopss  5067  frecxp  6314