Theorem eqsstrd 1534

Description: Substitution of equality into a subclass relationship.

Hypotheses

Ref	Expression
eqsstrd.1	|- (ph -> A = B)
eqsstrd.2	|- (ph -> B (_ C)

Assertion

Ref	Expression
eqsstrd	|- (ph -> A (_ C)

Proof of Theorem eqsstrd

Step	Hyp	Ref	Expression
1		eqsstrd.2	. 2 |- (ph -> B (_ C)
2		eqsstrd.1	. . 3 |- (ph -> A = B)
3	2	sseq1d 1527	. 2 |- (ph -> (A (_ C <-> B (_ C))
4	1, 3	mpbird 171	1 |- (ph -> A (_ C)

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

This theorem is referenced by:  eqsstr3d 1535  snsspr 1853  oawordeulem 3156  r1val1 3502  fodomb 3615  cardonle 3629  carduniima 3695  cfub 3703  cflecard 3707  infxpidmlem7 4939  infxpidmlem8 4940  atcvat3 5774

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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