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Theorem sbcom4 2445
Description: Commutativity law for substitution. This theorem was incorrectly used as our previous version of pm11.07 2446 but may still be useful. (Contributed by Andrew Salmon, 17-Jun-2011.) (Proof shortened by Jim Kingdon, 22-Jan-2018.)
Assertion
Ref Expression
sbcom4 ([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
Distinct variable groups:   𝜑,𝑥,𝑦,𝑧   𝑥,𝑤,𝑧
Allowed substitution hint:   𝜑(𝑤)

Proof of Theorem sbcom4
StepHypRef Expression
1 nfv 1840 . . 3 𝑥𝜑
21sbf 2379 . 2 ([𝑤 / 𝑥]𝜑𝜑)
3 nfv 1840 . . . 4 𝑧𝜑
43sbf 2379 . . 3 ([𝑦 / 𝑧]𝜑𝜑)
54sbbii 1884 . 2 ([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑤 / 𝑥]𝜑)
63sbf 2379 . . . 4 ([𝑤 / 𝑧]𝜑𝜑)
76sbbii 1884 . . 3 ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑)
81sbf 2379 . . 3 ([𝑦 / 𝑥]𝜑𝜑)
97, 8bitri 264 . 2 ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑𝜑)
102, 5, 93bitr4i 292 1 ([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsb 1877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707  df-sb 1878
This theorem is referenced by: (None)
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