Theorem psseq1 3031

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 2966	. . 3 |- ( A = B -> ( A C_ C <-> B C_ C ) )
2		neeq1 2218	. . 3 |- ( A = B -> ( A =/= C <-> B =/= C ) )
3	1, 2	anbi12d 442	. 2 |- ( A = B -> ( ( A C_ C /\ A =/= C ) <-> ( B C_ C /\ B =/= C ) ) )
4		df-pss 2933	. 2 |- ( A C. C <-> ( A C_ C /\ A =/= C ) )
5		df-pss 2933	. 2 |- ( B C. C <-> ( B C_ C /\ B =/= C ) )
6	3, 4, 5	3bitr4g 212	1 |- ( A = B -> ( A C. C <-> B C. C ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   =/= wne 2204   C_ wss 2917   C. wpss 2918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ne 2206  df-in 2924  df-ss 2931  df-pss 2933

This theorem is referenced by:  psseq1i  3033  psseq1d  3036  psstr  3049  psssstr  3051