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Theorem sbal 2127
 Description: Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbal ([z / y]xφx[z / y]φ)
Distinct variable groups:   x,y   x,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sbal
StepHypRef Expression
1 a16gb 2050 . . . . 5 (x x = z → (φxφ))
21sbimi 1652 . . . 4 ([z / y]x x = z → [z / y](φxφ))
3 sbequ5 2031 . . . 4 ([z / y]x x = zx x = z)
4 sbbi 2071 . . . 4 ([z / y](φxφ) ↔ ([z / y]φ ↔ [z / y]xφ))
52, 3, 43imtr3i 256 . . 3 (x x = z → ([z / y]φ ↔ [z / y]xφ))
6 a16gb 2050 . . 3 (x x = z → ([z / y]φx[z / y]φ))
75, 6bitr3d 246 . 2 (x x = z → ([z / y]xφx[z / y]φ))
8 sbal1 2126 . 2 x x = z → ([z / y]xφx[z / y]φ))
97, 8pm2.61i 156 1 ([z / y]xφx[z / y]φ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀wal 1540  [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:  sbex  2128  sbalv  2129  sbcal  3093  sbcalg  3094
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