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Theorem nfsb2or 1715
Description: Bound-variable hypothesis builder for substitution. Similar to hbsb2 1714 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 1711 . 2 (x x = y x([y / x]φx(x = yφ)))
2 sb2 1647 . . . . . . 7 (x(x = yφ) → [y / x]φ)
32a5i 1432 . . . . . 6 (x(x = yφ) → x[y / x]φ)
43imim2i 12 . . . . 5 (([y / x]φx(x = yφ)) → ([y / x]φx[y / x]φ))
54alimi 1341 . . . 4 (x([y / x]φx(x = yφ)) → x([y / x]φx[y / x]φ))
6 df-nf 1347 . . . 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 677 . 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 628  wal 1240  wnf 1346  [wsb 1642
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by:  sbequi  1717
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