Theorem sseq1d 2949

Description: An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)

Hypothesis

Ref	Expression
sseq1d.1	⊢ (φ → A = B)

Assertion

Ref	Expression
sseq1d	⊢ (φ → (A ⊆ 𝐶 ↔ B ⊆ 𝐶))

Proof of Theorem sseq1d

Step	Hyp	Ref	Expression
1		sseq1d.1	. 2 ⊢ (φ → A = B)
2		sseq1 2943	. 2 ⊢ (A = B → (A ⊆ 𝐶 ↔ B ⊆ 𝐶))
3	1, 2	syl 14	1 ⊢ (φ → (A ⊆ 𝐶 ↔ B ⊆ 𝐶))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1228   ⊆ wss 2894

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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-in 2901  df-ss 2908

This theorem is referenced by:  sseq12d  2951  eqsstrd  2956  snssg  3474  ssiun2s  3675  treq  3834  onsucsssucexmid  4196  funimass1  4902  feq1  4956  sbcfg  4971  fvmptssdm  5180  fvimacnvi  5206  nnsucsssuc  5986  ereq1  6024