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Theorem sbcom2 1863
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 1862 . . . 4 ([𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑣 / 𝑧]𝜑)
21sbbii 1648 . . 3 ([𝑤 / 𝑣][𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑧]𝜑)
3 sbcom2v2 1862 . . 3 ([𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑧]𝜑)
42, 3bitri 173 . 2 ([𝑤 / 𝑣][𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑧]𝜑)
5 ax-17 1419 . . 3 ([𝑦 / 𝑥]𝜑 → ∀𝑣[𝑦 / 𝑥]𝜑)
65sbco2v 1821 . 2 ([𝑤 / 𝑣][𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑)
7 ax-17 1419 . . . 4 (𝜑 → ∀𝑣𝜑)
87sbco2v 1821 . . 3 ([𝑤 / 𝑣][𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑)
98sbbii 1648 . 2 ([𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
104, 6, 93bitr3i 199 1 ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 98  [wsb 1645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646
This theorem is referenced by:  2sb5rf  1865  2sb6rf  1866  sbco4lem  1882  sbco4  1883  sbmo  1959  cnvopab  4726
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