Theorem hbss 2062

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 groups:   y,A   y,B   x,y

Proof of Theorem hbss

Step	Hyp	Ref	Expression
1		hba1 1003	. 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 2060	. 2 |- (A (_ B <-> A.x(x e. A -> x e. B))
5	4	albii 999	. 2 |- (A.x A (_ B <-> A.xA.x(x e. A -> x e. B))
6	1, 4, 5	3imtr4 219	1 |- (A (_ B -> A.x A (_ B)

Syntax hints:   -> wi 3  A.wal 954   e. wcel 958   (_ wss 2047

This theorem is referenced by:  hbpw 2407  ssiun2s 2594  ssopab2 2822  hbrel 3245  hbfun 3536  hbf 3625  rnssopab 3825  fopabco 3832  oawordeulem 4188  r1val1 4658  cardaleph 4885  tgval3t 7625  fgsb 10570  fgsbOLD 10571  fgsb2 10580

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-in 2051  df-ss 2053

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