Theorem 3sstr3 1538

Description: Substitution of equality in both sides of a subclass relationship.

Hypotheses

Ref	Expression
3sstr3.1	|- A (_ B
3sstr3.2	|- A = C
3sstr3.3	|- B = D

Assertion

Ref	Expression
3sstr3	|- C (_ D

Proof of Theorem 3sstr3

Step	Hyp	Ref	Expression
1		3sstr3.1	. 2 |- A (_ B
2		3sstr3.2	. . . 4 |- A = C
3	2	sseq1i 1524	. . 3 |- (A (_ B <-> C (_ B)
4		3sstr3.3	. . . 4 |- B = D
5	4	sseq2i 1525	. . 3 |- (C (_ B <-> C (_ D)
6	3, 5	bitr2 152	. 2 |- (C (_ D <-> A (_ B)
7	1, 6	mpbir 165	1 |- C (_ D

Syntax hints:   = wceq 1091   (_ wss 1487

This theorem is referenced by:  3sstr4 1539

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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