Theorem psseq1 2135

Description: Equality theorem for proper subclass.

Assertion

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

Proof of Theorem psseq1

Step	Hyp	Ref	Expression
1		sseq1 2082	. . 3 |- (A = B -> (A (_ C <-> B (_ C))
2		neeq1 1590	. . 3 |- (A = B -> (A =/= C <-> B =/= C))
3	1, 2	anbi12d 628	. 2 |- (A = B -> ((A (_ C /\ A =/= C) <-> (B (_ C /\ B =/= C)))
4		df-pss 2055	. 2 |- (A (. C <-> (A (_ C /\ A =/= C))
5		df-pss 2055	. 2 |- (B (. C <-> (B (_ C /\ B =/= C))
6	3, 4, 5	3bitr4g 555	1 |- (A = B -> (A (. C <-> B (. C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   =/= wne 1585   (_ wss 2047   (. wpss 2048

This theorem is referenced by:  psseq1i 2137  psseq1d 2140  ssnpss 2149  psstr 2150  sspsstr 2151  npss0 2309  pssnn 4534  infeq5 4621  zornlem 4795  elnp 5092  ltprord 5134  infxpidmlem10 7561  infpss 7574  spansncvt 9598  cvbrt 10209  cvcon3t 10211  cvnbtwnt 10213

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-in 2051  df-ss 2053  df-pss 2055

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