Theorem 3sstr4 1539

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

Hypotheses

Ref	Expression
3sstr4.1	|- A (_ B
3sstr4.2	|- C = A
3sstr4.3	|- D = B

Assertion

Ref	Expression
3sstr4	|- C (_ D

Proof of Theorem 3sstr4

Step	Hyp	Ref	Expression
1		3sstr4.1	. 2 |- A (_ B
2		3sstr4.2	. . 3 |- C = A
3	2	cleqcomi 1105	. 2 |- A = C
4		3sstr4.3	. . 3 |- D = B
5	4	cleqcomi 1105	. 2 |- B = D
6	1, 3, 5	3sstr3 1538	1 |- C (_ D

Syntax hints:   = wceq 1091   (_ wss 1487

This theorem is referenced by:  dmco 2570  rnco 2571  imassrn 2611  rnin 2645  ssoprab2i 3036  ranklon 3540  npex 3885  axresscn 4062  sshhococ 5451

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