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Theorem sbcom2 1963
 Description: Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.) (Proof modified to be intuitionistic by Jim Kingdon, 19-Feb-2018.)
Assertion
Ref Expression
sbcom2 ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
Distinct variable groups:   𝑥,𝑧   𝑥,𝑤   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem sbcom2
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 sbcom2v2 1962 . . . 4 ([𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑣 / 𝑧]𝜑)
21sbbii 1739 . . 3 ([𝑤 / 𝑣][𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑧]𝜑)
3 sbcom2v2 1962 . . 3 ([𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑧]𝜑)
42, 3bitri 183 . 2 ([𝑤 / 𝑣][𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑧]𝜑)
5 ax-17 1507 . . 3 ([𝑦 / 𝑥]𝜑 → ∀𝑣[𝑦 / 𝑥]𝜑)
65sbco2vh 1919 . 2 ([𝑤 / 𝑣][𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑)
7 ax-17 1507 . . . 4 (𝜑 → ∀𝑣𝜑)
87sbco2vh 1919 . . 3 ([𝑤 / 𝑣][𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑)
98sbbii 1739 . 2 ([𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
104, 6, 93bitr3i 209 1 ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104  [wsb 1736 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737 This theorem is referenced by:  2sb5rf  1965  2sb6rf  1966  sbco4lem  1982  sbco4  1983  sbmo  2059  cnvopab  4947
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