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Theorem sb4or 1711
 Description: One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1710 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 1708 . 2 (x x = y (x(x = y φ) → x(x = yφ)))
2 nfe1 1382 . . . . . 6 xx(x = y φ)
3 nfa1 1431 . . . . . 6 xx(x = yφ)
42, 3nfim 1461 . . . . 5 x(x(x = y φ) → x(x = yφ))
54nfri 1409 . . . 4 ((x(x = y φ) → x(x = yφ)) → x(x(x = y φ) → x(x = yφ)))
6 sb1 1646 . . . . 5 ([y / x]φx(x = y φ))
76imim1i 54 . . . 4 ((x(x = y φ) → x(x = yφ)) → ([y / x]φx(x = yφ)))
85, 7alrimih 1355 . . 3 ((x(x = y φ) → x(x = yφ)) → x([y / x]φx(x = yφ)))
98orim2i 677 . 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 628  ∀wal 1240  ∃wex 1378  [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:  sb4bor  1713  nfsb2or  1715
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