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Theorem sbex 2128
 Description: Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)
Assertion
Ref Expression
sbex ([z / y]xφx[z / y]φ)
Distinct variable groups:   x,y   x,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sbex
StepHypRef Expression
1 sbn 2062 . . 3 ([z / y] ¬ x ¬ φ ↔ ¬ [z / y]x ¬ φ)
2 sbal 2127 . . . 4 ([z / y]x ¬ φx[z / y] ¬ φ)
3 sbn 2062 . . . . 5 ([z / y] ¬ φ ↔ ¬ [z / y]φ)
43albii 1566 . . . 4 (x[z / y] ¬ φx ¬ [z / y]φ)
52, 4bitri 240 . . 3 ([z / y]x ¬ φx ¬ [z / y]φ)
61, 5xchbinx 301 . 2 ([z / y] ¬ x ¬ φ ↔ ¬ x ¬ [z / y]φ)
7 df-ex 1542 . . 3 (xφ ↔ ¬ x ¬ φ)
87sbbii 1653 . 2 ([z / y]xφ ↔ [z / y] ¬ x ¬ φ)
9 df-ex 1542 . 2 (x[z / y]φ ↔ ¬ x ¬ [z / y]φ)
106, 8, 93bitr4i 268 1 ([z / y]xφx[z / y]φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540  ∃wex 1541  [wsb 1648 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  sbmo  2234  sbabel  2515  sbcex2  3095  sbcexg  3096
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