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Theorem sbco2yz 1912
Description: This is a version of sbco2 1914 where 𝑧 is distinct from 𝑦. It is a lemma on the way to proving sbco2 1914 which has no distinct variable constraints. (Contributed by Jim Kingdon, 19-Mar-2018.)
Hypothesis
Ref Expression
sbco2yz.1 𝑧𝜑
Assertion
Ref Expression
sbco2yz ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbco2yz
StepHypRef Expression
1 sbco2yz.1 . . . 4 𝑧𝜑
21nfsb 1897 . . 3 𝑧[𝑦 / 𝑥]𝜑
32nfri 1482 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
4 sbequ 1794 . 2 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
53, 4sbieh 1746 1 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 104  wnf 1419  [wsb 1718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719
This theorem is referenced by:  sbco2h  1913
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