Theorem sseq12d 3301

Description: An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)

Hypotheses

Ref	Expression
sseq1d.1	⊢ (φ → A = B)
sseq12d.2	⊢ (φ → C = D)

Assertion

Ref	Expression
sseq12d	⊢ (φ → (A ⊆ C ↔ B ⊆ D))

Proof of Theorem sseq12d

Step	Hyp	Ref	Expression
1		sseq1d.1	. . 3 ⊢ (φ → A = B)
2	1	sseq1d 3299	. 2 ⊢ (φ → (A ⊆ C ↔ B ⊆ C))
3		sseq12d.2	. . 3 ⊢ (φ → C = D)
4	3	sseq2d 3300	. 2 ⊢ (φ → (B ⊆ C ↔ B ⊆ D))
5	2, 4	bitrd 244	1 ⊢ (φ → (A ⊆ C ↔ B ⊆ D))

Syntax hints:   → wi 4   ↔ 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:  3sstr3d  3314  3sstr4d  3315  ssdifeq0  3633  clos1induct  5881  frecxp  6315  frecxpg  6316