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Theorem sbco2v 1863
Description: This is a version of sbco2 1881 where 𝑧 is distinct from 𝑥. (Contributed by Jim Kingdon, 12-Feb-2018.)
Hypothesis
Ref Expression
sbco2v.1 (𝜑 → ∀𝑧𝜑)
Assertion
Ref Expression
sbco2v ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbco2v
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 sbco2v.1 . . . 4 (𝜑 → ∀𝑧𝜑)
21sbco2vlem 1862 . . 3 ([𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑)
32sbbii 1689 . 2 ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
4 ax-17 1460 . . 3 ([𝑧 / 𝑥]𝜑 → ∀𝑤[𝑧 / 𝑥]𝜑)
54sbco2vlem 1862 . 2 ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
6 ax-17 1460 . . 3 (𝜑 → ∀𝑤𝜑)
76sbco2vlem 1862 . 2 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
83, 5, 73bitr3i 208 1 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1283  [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by:  nfsb  1864  equsb3  1867  sbn  1868  sbim  1869  sbor  1870  sban  1871  sbco2vd  1883  sbco3v  1885  sbcom2v2  1904  sbcom2  1905  dfsb7  1909  sb7f  1910  sbal  1918  sbal1  1920  sbex  1922
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