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Theorem sbco2h 1976
Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.)
Hypothesis
Ref Expression
sbco2h.1 (𝜑 → ∀𝑧𝜑)
Assertion
Ref Expression
sbco2h ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Proof of Theorem sbco2h
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 sbco2h.1 . . . . 5 (𝜑 → ∀𝑧𝜑)
21nfi 1473 . . . 4 𝑧𝜑
32sbco2yz 1975 . . 3 ([𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑)
43sbbii 1776 . 2 ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
5 nfv 1539 . . 3 𝑤[𝑧 / 𝑥]𝜑
65sbco2yz 1975 . 2 ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
7 nfv 1539 . . 3 𝑤𝜑
87sbco2yz 1975 . 2 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
94, 6, 83bitr3i 210 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-bndl 1520  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:  sbco2  1977  sbco2d  1978  sbco3  1986  sb9  1991  elsb1  2167  elsb2  2168
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