Theorem eqsstrrid 3272

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

Hypotheses

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

Assertion

Ref	Expression
eqsstrrid	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem eqsstrrid

Step	Hyp	Ref	Expression
1		eqsstrrid.1	. . 3 ⊢ 𝐵 = 𝐴
2	1	eqcomi 2233	. 2 ⊢ 𝐴 = 𝐵
3		eqsstrrid.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
4	2, 3	eqsstrid 3271	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1395   ⊆ wss 3198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3204  df-ss 3211

This theorem is referenced by:  abnexg  4541  relcnvtr  5254  resasplitss  5513  fimacnvdisj  5518  fimacnv  5772  f1ompt  5794  tfr1onlemres  6510  tfrcllemres  6523