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Theorem sbco2h 1964
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 1462 . . . 4 𝑧𝜑
32sbco2yz 1963 . . 3 ([𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑)
43sbbii 1765 . 2 ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
5 nfv 1528 . . 3 𝑤[𝑧 / 𝑥]𝜑
65sbco2yz 1963 . 2 ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
7 nfv 1528 . . 3 𝑤𝜑
87sbco2yz 1963 . 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 1351  [wsb 1762
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763
This theorem is referenced by:  sbco2  1965  sbco2d  1966  sbco3  1974  sb9  1979  elsb1  2155  elsb2  2156
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