Theorem 3sstr3d 2106

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

Hypotheses

Ref	Expression
3sstr3d.1	|- (ph -> A (_ B)
3sstr3d.2	|- (ph -> A = C)
3sstr3d.3	|- (ph -> B = D)

Assertion

Ref	Expression
3sstr3d	|- (ph -> C (_ D)

Proof of Theorem 3sstr3d

Step	Hyp	Ref	Expression
1		3sstr3d.1	. 2 |- (ph -> A (_ B)
2		3sstr3d.2	. . 3 |- (ph -> A = C)
3		3sstr3d.3	. . 3 |- (ph -> B = D)
4	2, 3	sseq12d 2093	. 2 |- (ph -> (A (_ B <-> C (_ D))
5	1, 4	mpbid 195	1 |- (ph -> C (_ D)

Syntax hints:   -> wi 3   = wceq 958   (_ wss 2050

This theorem is referenced by:  shlej2t 9351  pjspansnt 9495

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-in 2054  df-ss 2056

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