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Theorem sb7af 1924
 Description: An alternate definition of proper substitution df-sb 1700. Similar to dfsb7a 1925 but does not require that 𝜑 and 𝑧 be distinct. Similar to sb7f 1923 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 1821 . . 3 ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧𝜑))
21sbbii 1702 . 2 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]∀𝑥(𝑥 = 𝑧𝜑))
3 sb7af.1 . . 3 𝑧𝜑
43sbco2 1894 . 2 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
5 sb6 1821 . 2 ([𝑦 / 𝑧]∀𝑥(𝑥 = 𝑧𝜑) ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)))
62, 4, 53bitr3i 209 1 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1294  Ⅎwnf 1401  [wsb 1699 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480 This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700 This theorem is referenced by:  dfsb7a  1925
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