Theorem sseq12 3294

Description: Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)

Assertion

Ref	Expression
sseq12	⊢ ((A = B ∧ C = D) → (A ⊆ C ↔ B ⊆ D))

Proof of Theorem sseq12

Step	Hyp	Ref	Expression
1		sseq1 3292	. 2 ⊢ (A = B → (A ⊆ C ↔ B ⊆ C))
2		sseq2 3293	. 2 ⊢ (C = D → (B ⊆ C ↔ B ⊆ D))
3	1, 2	sylan9bb 680	1 ⊢ ((A = B ∧ C = D) → (A ⊆ C ↔ B ⊆ D))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ⊆ wss 3257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by:  sseq12i  3297  ssfin  4470  funcnvuni  5161  fun11iun  5305