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Theorem sbal1 2126
Description: A theorem used in elimination of disjoint variable restriction on and by replacing it with a distinctor . (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbal1
Distinct variable group:   ,
Allowed substitution hints:   (,,)

Proof of Theorem sbal1
StepHypRef Expression
1 sbequ12 1919 . . . . 5
21sps 1754 . . . 4
3 sbequ12 1919 . . . . . 6
43sps 1754 . . . . 5
54dral2 1966 . . . 4
62, 5bitr3d 246 . . 3
76a1d 22 . 2
8 nfa1 1788 . . . . . . . 8  F/
98nfsb4 2081 . . . . . . 7  F/
109nfrd 1763 . . . . . 6
11 sp 1747 . . . . . . . 8
1211sbimi 1652 . . . . . . 7
1312alimi 1559 . . . . . 6
1410, 13syl6 29 . . . . 5
1514adantl 452 . . . 4
16 sb4 2053 . . . . . . . 8
1716al2imi 1561 . . . . . . 7
1817hbnaes 1957 . . . . . 6
19 ax-7 1734 . . . . . 6
2018, 19syl6 29 . . . . 5
21 dveeq2 1940 . . . . . . . . 9
22 alim 1558 . . . . . . . . 9
2321, 22syl9 66 . . . . . . . 8
2423al2imi 1561 . . . . . . 7
25 sb2 2023 . . . . . . 7
2624, 25syl6 29 . . . . . 6
2726hbnaes 1957 . . . . 5
2820, 27sylan9 638 . . . 4
2915, 28impbid 183 . . 3
3029ex 423 . 2
317, 30pm2.61i 156 1
Colors of variables: wff setvar class
Syntax hints:   wn 3   wi 4   wb 176   wa 358  wal 1540  wsb 1648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649
This theorem is referenced by:  sbal  2127
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