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Theorem sb4bor 1713
 Description: Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.)
Assertion
Ref Expression
sb4bor (x x = y x([y / x]φx(x = yφ)))

Proof of Theorem sb4bor
StepHypRef Expression
1 sb4or 1711 . 2 (x x = y x([y / x]φx(x = yφ)))
2 sb2 1647 . . . . 5 (x(x = yφ) → [y / x]φ)
3 df-bi 110 . . . . . 6 ((([y / x]φx(x = yφ)) → (([y / x]φx(x = yφ)) (x(x = yφ) → [y / x]φ))) ((([y / x]φx(x = yφ)) (x(x = yφ) → [y / x]φ)) → ([y / x]φx(x = yφ))))
43simpri 106 . . . . 5 ((([y / x]φx(x = yφ)) (x(x = yφ) → [y / x]φ)) → ([y / x]φx(x = yφ)))
52, 4mpan2 401 . . . 4 (([y / x]φx(x = yφ)) → ([y / x]φx(x = yφ)))
65alimi 1341 . . 3 (x([y / x]φx(x = yφ)) → x([y / x]φx(x = yφ)))
76orim2i 677 . 2 ((x x = y x([y / x]φx(x = yφ))) → (x x = y x([y / x]φx(x = yφ))))
81, 7ax-mp 7 1 (x x = y x([y / x]φx(x = yφ)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628  ∀wal 1240  [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: (None)
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