Theorem dfss3f 2057

Description: Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions.

Hypotheses

Ref	Expression
dfss2f.1	|- (y e. A -> A.x y e. A)
dfss2f.2	|- (y e. B -> A.x y e. B)

Assertion

Ref	Expression
dfss3f	|- (A (_ B <-> A.x e. A x e. B)

Distinct variable groups:   y,A   y,B   x,y

Proof of Theorem dfss3f

Step	Hyp	Ref	Expression
1		dfss2f.1	. . 3 |- (y e. A -> A.x y e. A)
2		dfss2f.2	. . 3 |- (y e. B -> A.x y e. B)
3	1, 2	dfss2f 2056	. 2 |- (A (_ B <-> A.x(x e. A -> x e. B))
4		df-ral 1646	. 2 |- (A.x e. A x e. B <-> A.x(x e. A -> x e. B))
5	3, 4	bitr4 176	1 |- (A (_ B <-> A.x e. A x e. B)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 952   e. wcel 956  A.wral 1642   (_ wss 2043

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-in 2047  df-ss 2049

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