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Theorem dfsb7a 1870
Description: An alternative definition of proper substitution df-sb 1646. Similar to dfsb7 1867 in that it involves a dummy variable 𝑧, but expressed in terms of rather than . For a version which only requires 𝑧𝜑 rather than 𝑧 and 𝜑 being distinct, see sb7af 1869. (Contributed by Jim Kingdon, 5-Feb-2018.)
Assertion
Ref Expression
dfsb7a ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem dfsb7a
StepHypRef Expression
1 nfv 1421 . 2 𝑧𝜑
21sb7af 1869 1 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1241  [wsb 1645
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646
This theorem is referenced by: (None)
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