Theorem sseq2d 3037

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
sseq2d	⊢ (𝜑 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵))

Proof of Theorem sseq2d

Step	Hyp	Ref	Expression
1		sseq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		sseq2 3031	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 103   = wceq 1285   ⊆ wss 2983

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-in 2989  df-ss 2996

This theorem is referenced by:  sseq12d  3038  sseqtrd  3045  onsucsssucexmid  4299  sbcrel  4473  funimass2  5029  fnco  5059  fnssresb  5063  fnimaeq0  5072  foimacnv  5196  fvelimab  5282  ssimaexg  5288  fvmptss2  5300  rdgss  6053