Theorem hbss 1501

Description: If x is not free in A and B, it is not free in A (_ B.

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
hbss	|- (A (_ B -> A.x A (_ B)

Distinct variable group(s):   y,A   y,B   x,y

Proof of Theorem hbss

Step	Hyp	Ref	Expression
1		hba1 698	. 2 |- (A.x(x e. A -> x e. B) -> A.xA.x(x e. A -> x e. B))
2		dfss2f.1	. . 3 |- (y e. A -> A.x y e. A)
3		dfss2f.2	. . 3 |- (y e. B -> A.x y e. B)
4	2, 3	dfss2f 1499	. 2 |- (A (_ B <-> A.x(x e. A -> x e. B))
5	4	bial 695	. 2 |- (A.x A (_ B <-> A.xA.x(x e. A -> x e. B))
6	1, 4, 5	3imtr4 192	1 |- (A (_ B -> A.x A (_ B)

Syntax hints:   -> wi 2  A.wal 672   e. wcel 1092   (_ wss 1487

This theorem is referenced by:  hbpw 1804  ssiun2s 2020  ssopab2 2119  hbrel 2478  hbfun 2684  hbf 2751  oawordeulem 3156  r1val1 3502  cardaleph 3690

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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