Theorem psseq1 3356

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

Assertion

Ref	Expression
psseq1	⊢ (A = B → (A ⊊ C ↔ B ⊊ C))

Proof of Theorem psseq1

Step	Hyp	Ref	Expression
1		sseq1 3292	. . 3 ⊢ (A = B → (A ⊆ C ↔ B ⊆ C))
2		neeq1 2524	. . 3 ⊢ (A = B → (A ≠ C ↔ B ≠ C))
3	1, 2	anbi12d 691	. 2 ⊢ (A = B → ((A ⊆ C ∧ A ≠ C) ↔ (B ⊆ C ∧ B ≠ C)))
4		df-pss 3261	. 2 ⊢ (A ⊊ C ↔ (A ⊆ C ∧ A ≠ C))
5		df-pss 3261	. 2 ⊢ (B ⊊ C ↔ (B ⊆ C ∧ B ≠ C))
6	3, 4, 5	3bitr4g 279	1 ⊢ (A = B → (A ⊊ C ↔ B ⊊ C))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ≠ wne 2516   ⊆ wss 3257   ⊊ wpss 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 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pss 3261

This theorem is referenced by:  psseq1i  3358  psseq1d  3361  psstr  3373  sspsstr  3374