Theorem sseq12i 1526

Description: An equality inference for the subclass relationship.

Hypotheses

Ref	Expression
sseq1i.1	|- A = B
sseq12i.2	|- C = D

Assertion

Ref	Expression
sseq12i	|- (A (_ C <-> B (_ D)

Proof of Theorem sseq12i

Step	Hyp	Ref	Expression
1		sseq1i.1	. . 3 |- A = B
2	1	sseq1i 1524	. 2 |- (A (_ C <-> B (_ C)
3		sseq12i.2	. . 3 |- C = D
4	3	sseq2i 1525	. 2 |- (B (_ C <-> B (_ D)
5	2, 4	bitr 151	1 |- (A (_ C <-> B (_ D)

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

This theorem is referenced by:  3sstr3g 1540  ss2rab 1553  ssopab2 2119  shlub 5347  pjoi0 5592

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

metamath.org