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Theorem sbcrot3 39266
Description: Rotate a sequence of three explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
sbcrot3 ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐]𝜑[𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎]𝜑)
Distinct variable groups:   𝐴,𝑏   𝐴,𝑐   𝐵,𝑎   𝐶,𝑎   𝑎,𝑐   𝑎,𝑏
Allowed substitution hints:   𝜑(𝑎,𝑏,𝑐)   𝐴(𝑎)   𝐵(𝑏,𝑐)   𝐶(𝑏,𝑐)

Proof of Theorem sbcrot3
StepHypRef Expression
1 sbccom 3851 . 2 ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐]𝜑[𝐵 / 𝑏][𝐴 / 𝑎][𝐶 / 𝑐]𝜑)
2 sbccom 3851 . . 3 ([𝐴 / 𝑎][𝐶 / 𝑐]𝜑[𝐶 / 𝑐][𝐴 / 𝑎]𝜑)
32sbcbii 3826 . 2 ([𝐵 / 𝑏][𝐴 / 𝑎][𝐶 / 𝑐]𝜑[𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎]𝜑)
41, 3bitri 276 1 ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐]𝜑[𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 207  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-v 3494  df-sbc 3770
This theorem is referenced by:  sbcrot5  39267  2rexfrabdioph  39271  3rexfrabdioph  39272  4rexfrabdioph  39273  7rexfrabdioph  39275
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