Theorem sseq12i 3298

Description: An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
sseq1i.1	⊢ A = B
sseq12i.2	⊢ C = D

Assertion

Ref	Expression
sseq12i	⊢ (A ⊆ C ↔ B ⊆ D)

Proof of Theorem sseq12i

Step	Hyp	Ref	Expression
1		sseq1i.1	. 2 ⊢ A = B
2		sseq12i.2	. 2 ⊢ C = D
3		sseq12 3295	. 2 ⊢ ((A = B ∧ C = D) → (A ⊆ C ↔ B ⊆ D))
4	1, 2, 3	mp2an 653	1 ⊢ (A ⊆ C ↔ B ⊆ D)

Syntax hints:   ↔ wb 176   = wceq 1642   ⊆ wss 3258

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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  3sstr3i  3310  3sstr4i  3311  3sstr3g  3312  3sstr4g  3313  ss2rab  3343