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Theorem sb4or 1692
Description: One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1691 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.)
Assertion
Ref Expression
sb4or (x x = y x([y / x]φx(x = yφ)))

Proof of Theorem sb4or
StepHypRef Expression
1 equs5or 1689 . 2 (x x = y (x(x = y φ) → x(x = yφ)))
2 nfe1 1362 . . . . . 6 xx(x = y φ)
3 nfa1 1412 . . . . . 6 xx(x = yφ)
42, 3nfim 1442 . . . . 5 x(x(x = y φ) → x(x = yφ))
54nfri 1389 . . . 4 ((x(x = y φ) → x(x = yφ)) → x(x(x = y φ) → x(x = yφ)))
6 sb1 1627 . . . . 5 ([y / x]φx(x = y φ))
76imim1i 54 . . . 4 ((x(x = y φ) → x(x = yφ)) → ([y / x]φx(x = yφ)))
85, 7alrimih 1334 . . 3 ((x(x = y φ) → x(x = yφ)) → x([y / x]φx(x = yφ)))
98orim2i 665 . 2 ((x x = y (x(x = y φ) → x(x = yφ))) → (x x = y x([y / x]φx(x = yφ))))
101, 9ax-mp 7 1 (x x = y x([y / x]φx(x = yφ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 616  wal 1224  wex 1358  [wsb 1623
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624
This theorem is referenced by:  sb4bor  1694  nfsb2or  1696
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