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Theorem sbcom4 2100
 Description: Commutativity law for substitution. This theorem was incorrectly used as our previous version of pm11.07 2101 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   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑤)

Proof of Theorem sbcom4
StepHypRef Expression
1 sbv 2099 . 2 ([𝑤 / 𝑥]𝜑𝜑)
2 sbv 2099 . . 3 ([𝑦 / 𝑧]𝜑𝜑)
32sbbii 2082 . 2 ([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑤 / 𝑥]𝜑)
4 sbv 2099 . . . 4 ([𝑤 / 𝑧]𝜑𝜑)
54sbbii 2082 . . 3 ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑)
6 sbv 2099 . . 3 ([𝑦 / 𝑥]𝜑𝜑)
75, 6bitri 278 . 2 ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑𝜑)
81, 3, 73bitr4i 306 1 ([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  [wsb 2070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971 This theorem depends on definitions:  df-bi 210  df-ex 1782  df-sb 2071 This theorem is referenced by: (None)
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