Theorem sseq1i 3230

Description: An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)

Hypothesis

Ref	Expression
sseq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
sseq1i	⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)

Proof of Theorem sseq1i

Step	Hyp	Ref	Expression
1		sseq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		sseq1 3227	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)

Syntax hints:   ↔ wb 105   = wceq 1375   ⊆ wss 3177

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 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-11 1532  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-in 3183  df-ss 3190

This theorem is referenced by:  eqsstri  3236  eqsstrid  3250  ssab  3274  rabss  3281  uniiunlem  3293  prss  3803  prssg  3804  tpss  3815  iunss  3985  pwtr  4284  ordsucss  4573  elomssom  4674  cores2  5217  dffun2  5304  funimaexglem  5380  idref  5853  ordgt0ge1  6551  3nsssucpw1  7389  prarloclemn  7654  ausgrusgrben  15931  bdeqsuc  16154  bj-omssind  16208