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Theorem sb7af 1885
Description: An alternative definition of proper substitution df-sb 1662. Similar to dfsb7a 1886 but does not require that 𝜑 and 𝑧 be distinct. Similar to sb7f 1884 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]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sb7af
StepHypRef Expression
1 sb6 1782 . . 3 ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧𝜑))
21sbbii 1664 . 2 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]∀𝑥(𝑥 = 𝑧𝜑))
3 sb7af.1 . . 3 𝑧𝜑
43sbco2 1855 . 2 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
5 sb6 1782 . 2 ([𝑦 / 𝑧]∀𝑥(𝑥 = 𝑧𝜑) ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)))
62, 4, 53bitr3i 203 1 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257  wnf 1365  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662
This theorem is referenced by:  dfsb7a  1886
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