Theorem psseq1 3357

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

Assertion

Ref	Expression
psseq1	|- (A = B -> (A C. C <-> B C. C))

Proof of Theorem psseq1

Step	Hyp	Ref	Expression
1		sseq1 3293	. . 3 |- (A = B -> (A C_ C <-> B C_ C))
2		neeq1 2525	. . 3 |- (A = B -> (A =/= C <-> B =/= C))
3	1, 2	anbi12d 691	. 2 |- (A = B -> ((A C_ C /\ A =/= C) <-> (B C_ C /\ B =/= C)))
4		df-pss 3262	. 2 |- (A C. C <-> (A C_ C /\ A =/= C))
5		df-pss 3262	. 2 |- (B C. C <-> (B C_ C /\ B =/= C))
6	3, 4, 5	3bitr4g 279	1 |- (A = B -> (A C. C <-> B C. C))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   =/= wne 2517   C_ wss 3258   C. 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:  psseq1i  3359  psseq1d  3362  psstr  3374  sspsstr  3375