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Theorem sbequi 2112
 Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Jul-2018.)
Assertion
Ref Expression
sbequi

Proof of Theorem sbequi
StepHypRef Expression
1 equvini 2084 . . . . 5
2 sbequ2 1661 . . . . . . . . 9
3 sbequ1 1944 . . . . . . . . 9
42, 3syl9 69 . . . . . . . 8
54equcoms 1694 . . . . . . 7
65imp 420 . . . . . 6
76eximi 1586 . . . . 5
81, 7syl 16 . . . 4
9 nfsb2 2098 . . . . . . 7
109adantr 453 . . . . . 6
11 nfsb2 2098 . . . . . . 7
1211adantl 454 . . . . . 6
1310, 12nfimd 1828 . . . . 5
141319.9d 1797 . . . 4
158, 14syl5 31 . . 3
1615ex 425 . 2
17 equtr 1695 . . . 4
1817, 4syld 43 . . 3
1918sps 1771 . 2
20 equequ2 1699 . . . . . 6
2120biimprd 216 . . . . 5
2221, 4syli 36 . . . 4
2322com12 30 . . 3
2423sps 1771 . 2
2516, 19, 24pm2.61ii 160 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 360  wal 1550  wex 1551  wnf 1554  wsb 1659 This theorem is referenced by:  sbequ  2113 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951 This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660
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