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Theorem sbcom2 2114
Description: Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.)
Assertion
Ref Expression
sbcom2
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   (,,,)

Proof of Theorem sbcom2
StepHypRef Expression
1 alcom 1737 . . . . . 6
2 bi2.04 350 . . . . . . . . 9
32albii 1566 . . . . . . . 8
4 19.21v 1890 . . . . . . . 8
53, 4bitri 240 . . . . . . 7
65albii 1566 . . . . . 6
7 19.21v 1890 . . . . . . 7
87albii 1566 . . . . . 6
91, 6, 83bitr3i 266 . . . . 5
109a1i 10 . . . 4
11 sb4b 2054 . . . . 5
12 sb4b 2054 . . . . . . 7
1312imbi2d 307 . . . . . 6
1413albidv 1625 . . . . 5
1511, 14sylan9bbr 681 . . . 4
16 sb4b 2054 . . . . 5
17 sb4b 2054 . . . . . . 7
1817imbi2d 307 . . . . . 6
1918albidv 1625 . . . . 5
2016, 19sylan9bb 680 . . . 4
2110, 15, 203bitr4d 276 . . 3
2221ex 423 . 2
23 nfae 1954 . . . 4  F/
24 sbequ12 1919 . . . . 5
2524sps 1754 . . . 4
2623, 25sbbid 2078 . . 3
27 sbequ12 1919 . . . 4
2827sps 1754 . . 3
2926, 28bitr3d 246 . 2
30 sbequ12 1919 . . . 4
3130sps 1754 . . 3
32 nfae 1954 . . . 4  F/
33 sbequ12 1919 . . . . 5
3433sps 1754 . . . 4
3532, 34sbbid 2078 . . 3
3631, 35bitr3d 246 . 2
3722, 29, 36pm2.61ii 157 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:  2sb5rf  2117  2sb6rf  2118  2eu6  2289
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