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Theorem sbco2vlem 1956
Description: This is a version of sbco2 1977 where 𝑧 is distinct from 𝑥 and from 𝑦. It is a lemma on the way to proving sbco2v 1960 which only requires that 𝑧 and 𝑥 be distinct. (Contributed by Jim Kingdon, 25-Dec-2017.) Remove one disjoint variable condition. (Revised by Jim Kingdon, 3-Feb-2018.)
Hypothesis
Ref Expression
sbco2vlem.1 (𝜑 → ∀𝑧𝜑)
Assertion
Ref Expression
sbco2vlem ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbco2vlem
StepHypRef Expression
1 sbco2vlem.1 . . 3 (𝜑 → ∀𝑧𝜑)
21hbsbv 1953 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
3 sbequ 1851 . 2 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
42, 3sbieh 1801 1 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1362  [wsb 1773
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774
This theorem is referenced by:  sbco2vh  1957
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