Theorem sseq1i 3154

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 3151	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)

Syntax hints:   ↔ wb 104   = wceq 1335   ⊆ wss 3102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-in 3108  df-ss 3115

This theorem is referenced by:  eqsstri  3160  eqsstrid  3174  ssab  3198  rabss  3205  uniiunlem  3216  prss  3712  prssg  3713  tpss  3721  iunss  3890  pwtr  4178  ordsucss  4461  elomssom  4562  cores2  5095  dffun2  5177  funimaexglem  5250  idref  5702  ordgt0ge1  6376  3nsssucpw1  7154  prarloclemn  7402  bdeqsuc  13416  bj-omssind  13470