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Theorem nfsb2or 1696
Description: Bound-variable hypothesis builder for substitution. Similar to hbsb2 1695 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.)
Assertion
Ref Expression
nfsb2or (x x = y x[y / x]φ)

Proof of Theorem nfsb2or
StepHypRef Expression
1 sb4or 1692 . 2 (x x = y x([y / x]φx(x = yφ)))
2 sb2 1628 . . . . . . 7 (x(x = yφ) → [y / x]φ)
32a5i 1413 . . . . . 6 (x(x = yφ) → x[y / x]φ)
43imim2i 12 . . . . 5 (([y / x]φx(x = yφ)) → ([y / x]φx[y / x]φ))
54alimi 1320 . . . 4 (x([y / x]φx(x = yφ)) → x([y / x]φx[y / x]φ))
6 df-nf 1326 . . . 4 (Ⅎx[y / x]φx([y / x]φx[y / x]φ))
75, 6sylibr 137 . . 3 (x([y / x]φx(x = yφ)) → Ⅎx[y / x]φ)
87orim2i 665 . 2 ((x x = y x([y / x]φx(x = yφ))) → (x x = y x[y / x]φ))
91, 8ax-mp 7 1 (x x = y x[y / x]φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wo 616  wal 1224  wnf 1325  [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:  sbequi  1698
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