Theorem sseq2i 3211

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 3208	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵)

Syntax hints:   ↔ wb 105   = wceq 1364   ⊆ wss 3157

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170

This theorem is referenced by:  sseqtri  3218  sseqtrdi  3232  abss  3253  ssrab  3262  ssintrab  3898  iunpwss  4009  iotass  5237  dffun2  5269  ssimaex  5625  pw1fin  6980  pw1dc0el  6981  ss1o0el1o  6983  isstructim  12717  isstructr  12718  issubm  13174  grpissubg  13400  issubrng  13831  bj-ssom  15666  ss1oel2o  15722