Theorem sseq2i 3251

Description: An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)

Hypothesis

Ref	Expression
sseq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
sseq2i	⊢ (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵)

Proof of Theorem sseq2i

Step	Hyp	Ref	Expression
1		sseq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		sseq2 3248	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵)

Syntax hints:   ↔ wb 105   = wceq 1395   ⊆ wss 3197

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 3203  df-ss 3210

This theorem is referenced by:  sseqtri  3258  sseqtrdi  3272  abss  3293  ssrab  3302  ssintrab  3945  iunpwss  4056  iotass  5295  dffun2  5327  ssimaex  5694  pw1fin  7068  pw1dc0el  7069  ss1o0el1o  7071  isstructim  13041  isstructr  13042  issubm  13500  grpissubg  13726  issubrng  14157  umgredg  15937  bj-ssom  16257  ss1oel2o  16313