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Theorem sb7af 1866
 Description: An alternative definition of proper substitution df-sb 1643. Similar to dfsb7a 1867 but does not require that φ and z be distinct. Similar to sb7f 1865 in that it involves a dummy variable z, but expressed in terms of ∀ rather than ∃. (Contributed by Jim Kingdon, 5-Feb-2018.)
Hypothesis
Ref Expression
sb7af.1 zφ
Assertion
Ref Expression
sb7af ([y / x]φz(z = yx(x = zφ)))
Distinct variable groups:   x,z   y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sb7af
StepHypRef Expression
1 sb6 1763 . . 3 ([z / x]φx(x = zφ))
21sbbii 1645 . 2 ([y / z][z / x]φ ↔ [y / z]x(x = zφ))
3 sb7af.1 . . 3 zφ
43sbco2 1836 . 2 ([y / z][z / x]φ ↔ [y / x]φ)
5 sb6 1763 . 2 ([y / z]x(x = zφ) ↔ z(z = yx(x = zφ)))
62, 4, 53bitr3i 199 1 ([y / x]φz(z = yx(x = zφ)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240  Ⅎwnf 1346  [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-bnd 1396  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:  dfsb7a  1867
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