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Theorem sbcrot3 39729
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 3803 . 2 ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐]𝜑[𝐵 / 𝑏][𝐴 / 𝑎][𝐶 / 𝑐]𝜑)
2 sbccom 3803 . . 3 ([𝐴 / 𝑎][𝐶 / 𝑐]𝜑[𝐶 / 𝑐][𝐴 / 𝑎]𝜑)
32sbcbii 3779 . 2 ([𝐵 / 𝑏][𝐴 / 𝑎][𝐶 / 𝑐]𝜑[𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎]𝜑)
41, 3bitri 278 1 ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐]𝜑[𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209  [wsbc 3723
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-sbc 3724
This theorem is referenced by:  sbcrot5  39730  2rexfrabdioph  39734  3rexfrabdioph  39735  4rexfrabdioph  39736  7rexfrabdioph  39738
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