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Theorem sbcom4 2537
Description: Commutativity law for substitution. This theorem was incorrectly used as our previous version of pm11.07 2538 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 2009 . . 3 𝑥𝜑
21sbf 2468 . 2 ([𝑤 / 𝑥]𝜑𝜑)
3 nfv 2009 . . . 4 𝑧𝜑
43sbf 2468 . . 3 ([𝑦 / 𝑧]𝜑𝜑)
54sbbii 2068 . 2 ([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑤 / 𝑥]𝜑)
63sbf 2468 . . . 4 ([𝑤 / 𝑧]𝜑𝜑)
76sbbii 2068 . . 3 ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑)
81sbf 2468 . . 3 ([𝑦 / 𝑥]𝜑𝜑)
97, 8bitri 266 . 2 ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑𝜑)
102, 5, 93bitr4i 294 1 ([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 197  [wsb 2061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-12 2211  ax-13 2349
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875  df-nf 1879  df-sb 2062
This theorem is referenced by: (None)
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