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Theorem sb7af 1993
Description: An alternate definition of proper substitution df-sb 1763. Similar to dfsb7a 1994 but does not require that 𝜑 and 𝑧 be distinct. Similar to sb7f 1992 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 1886 . . 3 ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧𝜑))
21sbbii 1765 . 2 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]∀𝑥(𝑥 = 𝑧𝜑))
3 sb7af.1 . . 3 𝑧𝜑
43sbco2 1965 . 2 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
5 sb6 1886 . 2 ([𝑦 / 𝑧]∀𝑥(𝑥 = 𝑧𝜑) ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)))
62, 4, 53bitr3i 210 1 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351  wnf 1460  [wsb 1762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763
This theorem is referenced by:  dfsb7a  1994
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