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Theorem sbcrot5 37673
Description: Rotate a sequence of five explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
sbcrot5 ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒]𝜑[𝐵 / 𝑏][𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑)
Distinct variable groups:   𝐴,𝑏   𝐴,𝑐   𝐵,𝑎   𝐶,𝑎   𝑎,𝑐   𝑎,𝑏   𝐴,𝑑   𝐴,𝑒   𝐷,𝑎   𝐸,𝑎   𝑒,𝑎   𝑎,𝑑
Allowed substitution hints:   𝜑(𝑒,𝑎,𝑏,𝑐,𝑑)   𝐴(𝑎)   𝐵(𝑒,𝑏,𝑐,𝑑)   𝐶(𝑒,𝑏,𝑐,𝑑)   𝐷(𝑒,𝑏,𝑐,𝑑)   𝐸(𝑒,𝑏,𝑐,𝑑)

Proof of Theorem sbcrot5
StepHypRef Expression
1 sbcrot3 37672 . 2 ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒]𝜑[𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎][𝐷 / 𝑑][𝐸 / 𝑒]𝜑)
2 sbcrot3 37672 . . . 4 ([𝐴 / 𝑎][𝐷 / 𝑑][𝐸 / 𝑒]𝜑[𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑)
32sbcbii 3524 . . 3 ([𝐶 / 𝑐][𝐴 / 𝑎][𝐷 / 𝑑][𝐸 / 𝑒]𝜑[𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑)
43sbcbii 3524 . 2 ([𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎][𝐷 / 𝑑][𝐸 / 𝑒]𝜑[𝐵 / 𝑏][𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑)
51, 4bitri 264 1 ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒]𝜑[𝐵 / 𝑏][𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsbc 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233  df-sbc 3469
This theorem is referenced by:  6rexfrabdioph  37680  7rexfrabdioph  37681
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