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Theorem sb7f 2120
 Description: This version of dfsb7 2119 does not require that φ and z be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1616 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1649 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypothesis
Ref Expression
sb7f.1 zφ
Assertion
Ref Expression
sb7f ([y / x]φz(z = y x(x = z φ)))
Distinct variable groups:   x,z   y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sb7f
StepHypRef Expression
1 sb5 2100 . . 3 ([z / x]φx(x = z φ))
21sbbii 1653 . 2 ([y / z][z / x]φ ↔ [y / z]x(x = z φ))
3 sb7f.1 . . 3 zφ
43sbco2 2086 . 2 ([y / z][z / x]φ ↔ [y / x]φ)
5 sb5 2100 . 2 ([y / z]x(x = z φ) ↔ z(z = y x(x = z φ)))
62, 4, 53bitr3i 266 1 ([y / x]φz(z = y x(x = z φ)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  sb7h  2121
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