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Theorem sb7f 1865
 Description: This version of dfsb7 1864 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 1416 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1643 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
sb7f.1 (φzφ)
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 1764 . . 3 ([z / x]φx(x = z φ))
21sbbii 1645 . 2 ([y / z][z / x]φ ↔ [y / z]x(x = z φ))
3 sb7f.1 . . 3 (φzφ)
43sbco2v 1818 . 2 ([y / z][z / x]φ ↔ [y / x]φ)
5 sb5 1764 . 2 ([y / z]x(x = z φ) ↔ z(z = y x(x = z φ)))
62, 4, 53bitr3i 199 1 ([y / x]φz(z = y x(x = z φ)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  ∃wex 1378  [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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