Theorem sseq12d 1529

Description: An equality deduction for the subclass relationship.

Hypotheses

Ref	Expression
sseq1d.1	|- (ph -> A = B)
sseq12d.2	|- (ph -> C = D)

Assertion

Ref	Expression
sseq12d	|- (ph -> (A (_ C <-> B (_ D))

Proof of Theorem sseq12d

Step	Hyp	Ref	Expression
1		sseq1d.1	. . 3 |- (ph -> A = B)
2	1	sseq1d 1527	. 2 |- (ph -> (A (_ C <-> B (_ C))
3		sseq12d.2	. . 3 |- (ph -> C = D)
4	3	sseq2d 1528	. 2 |- (ph -> (B (_ C <-> B (_ D))
5	2, 4	bitrd 406	1 |- (ph -> (A (_ C <-> B (_ D))

Syntax hints:   -> wi 2   <-> wb 127   = wceq 1091   (_ wss 1487

This theorem is referenced by:  3sstr3d 1542  tz6.12-2 2845  oawordri 3152  inf3lem1 3464  alephle 3689  hsupss 5310  shslejt 5351  osum 5538  mdbr4 5730

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492

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