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Theorem sb8e 2093
 Description: Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypothesis
Ref Expression
sb5rf.1 yφ
Assertion
Ref Expression
sb8e (xφy[y / x]φ)

Proof of Theorem sb8e
StepHypRef Expression
1 sb5rf.1 . . . . . 6 yφ
21nfn 1793 . . . . 5 y ¬ φ
32sb8 2092 . . . 4 (x ¬ φy[y / x] ¬ φ)
4 sbn 2062 . . . . 5 ([y / x] ¬ φ ↔ ¬ [y / x]φ)
54albii 1566 . . . 4 (y[y / x] ¬ φy ¬ [y / x]φ)
63, 5bitri 240 . . 3 (x ¬ φy ¬ [y / x]φ)
76notbii 287 . 2 x ¬ φ ↔ ¬ y ¬ [y / x]φ)
8 df-ex 1542 . 2 (xφ ↔ ¬ x ¬ φ)
9 df-ex 1542 . 2 (y[y / x]φ ↔ ¬ y ¬ [y / x]φ)
107, 8, 93bitr4i 268 1 (xφy[y / x]φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544  [wsb 1648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  exsbOLD  2131  sb8mo  2223
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