Theorem sseq2i 2964

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 2961	. 2 ⊢ (A = B → (𝐶 ⊆ A ↔ 𝐶 ⊆ B))
3	1, 2	ax-mp 7	1 ⊢ (𝐶 ⊆ A ↔ 𝐶 ⊆ B)

Syntax hints:   ↔ wb 98   = wceq 1242   ⊆ wss 2911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925

This theorem is referenced by:  sseqtri  2971  syl6sseq  2985  abss  3003  ssrab  3012  ssintrab  3629  iunpwss  3734  iotass  4827  dffun2  4855  ssimaex  5177  bj-ssom  9395