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Theorem elsb3 1834
Description: Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
elsb3 ([x / y]y zx z)
Distinct variable group:   y,z

Proof of Theorem elsb3
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 ax-17 1400 . . . . 5 (y zw y z)
2 elequ1 1582 . . . . 5 (w = y → (w zy z))
31, 2sbieh 1655 . . . 4 ([y / w]w zy z)
43sbbii 1630 . . 3 ([x / y][y / w]w z ↔ [x / y]y z)
5 ax-17 1400 . . . 4 (w zy w z)
65sbco2h 1820 . . 3 ([x / y][y / w]w z ↔ [x / w]w z)
74, 6bitr3i 175 . 2 ([x / y]y z ↔ [x / w]w z)
8 equsb1 1650 . . . 4 [x / w]w = x
9 elequ1 1582 . . . . 5 (w = x → (w zx z))
109sbimi 1629 . . . 4 ([x / w]w = x → [x / w](w zx z))
118, 10ax-mp 7 . . 3 [x / w](w zx z)
12 sbbi 1815 . . 3 ([x / w](w zx z) ↔ ([x / w]w z ↔ [x / w]x z))
1311, 12mpbi 133 . 2 ([x / w]w z ↔ [x / w]x z)
14 ax-17 1400 . . 3 (x zw x z)
1514sbh 1641 . 2 ([x / w]x zx z)
167, 13, 153bitri 195 1 ([x / y]y zx z)
Colors of variables: wff set class
Syntax hints:  wb 98  [wsb 1627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628
This theorem is referenced by:  cvjust  2017
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