Theorem psseq2 3358

Description: Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)

Assertion

Ref	Expression
psseq2	⊢ (A = B → (C ⊊ A ↔ C ⊊ B))

Proof of Theorem psseq2

Step	Hyp	Ref	Expression
1		sseq2 3294	. . 3 ⊢ (A = B → (C ⊆ A ↔ C ⊆ B))
2		neeq2 2526	. . 3 ⊢ (A = B → (C ≠ A ↔ C ≠ B))
3	1, 2	anbi12d 691	. 2 ⊢ (A = B → ((C ⊆ A ∧ C ≠ A) ↔ (C ⊆ B ∧ C ≠ B)))
4		df-pss 3262	. 2 ⊢ (C ⊊ A ↔ (C ⊆ A ∧ C ≠ A))
5		df-pss 3262	. 2 ⊢ (C ⊊ B ↔ (C ⊆ B ∧ C ≠ B))
6	3, 4, 5	3bitr4g 279	1 ⊢ (A = B → (C ⊊ A ↔ C ⊊ B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ≠ wne 2517   ⊆ wss 3258   ⊊ wpss 3259

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-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pss 3262

This theorem is referenced by:  psseq2i  3360  psseq2d  3363  psssstr  3376