Theorem sbcrot5 37356

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 ]. ph <-> [. B / b ]. [. C / c ]. [. D / d ]. [. E / e ]. [. A / a ]. ph )

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:   ph( 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

Step	Hyp	Ref	Expression
1		sbcrot3 37355	. 2 |- ( [. A / a ]. [. B / b ]. [. C / c ]. [. D / d ]. [. E / e ]. ph <-> [. B / b ]. [. C / c ]. [. A / a ]. [. D / d ]. [. E / e ]. ph )
2		sbcrot3 37355	. . . 4 |- ( [. A / a ]. [. D / d ]. [. E / e ]. ph <-> [. D / d ]. [. E / e ]. [. A / a ]. ph )
3	2	sbcbii 3491	. . 3 |- ( [. C / c ]. [. A / a ]. [. D / d ]. [. E / e ]. ph <-> [. C / c ]. [. D / d ]. [. E / e ]. [. A / a ]. ph )
4	3	sbcbii 3491	. 2 |- ( [. B / b ]. [. C / c ]. [. A / a ]. [. D / d ]. [. E / e ]. ph <-> [. B / b ]. [. C / c ]. [. D / d ]. [. E / e ]. [. A / a ]. ph )
5	1, 4	bitri 264	1 |- ( [. A / a ]. [. B / b ]. [. C / c ]. [. D / d ]. [. E / e ]. ph <-> [. B / b ]. [. C / c ]. [. D / d ]. [. E / e ]. [. A / a ]. ph )

Syntax hints:   <-> wb 196   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  6rexfrabdioph  37363  7rexfrabdioph  37364