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Theorem sbco2h 1937
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 1438 . . . 4 𝑧𝜑
32sbco2yz 1936 . . 3 ([𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑)
43sbbii 1738 . 2 ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
5 nfv 1508 . . 3 𝑤[𝑧 / 𝑥]𝜑
65sbco2yz 1936 . 2 ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
7 nfv 1508 . . 3 𝑤𝜑
87sbco2yz 1936 . 2 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
94, 6, 83bitr3i 209 1 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1329  [wsb 1735
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736
This theorem is referenced by:  sbco2  1938  sbco2d  1939  sbco3  1947  elsb3  1951  elsb4  1952  sb9  1954
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