Theorem sbcrot5 38142

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

Step	Hyp	Ref	Expression
1		sbcrot3 38141	. 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 38141	. . . 4 ⊢ ([𝐴 / 𝑎][𝐷 / 𝑑][𝐸 / 𝑒]𝜑 ↔ [𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑)
3	2	sbcbii 3689	. . 3 ⊢ ([𝐶 / 𝑐][𝐴 / 𝑎][𝐷 / 𝑑][𝐸 / 𝑒]𝜑 ↔ [𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑)
4	3	sbcbii 3689	. 2 ⊢ ([𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎][𝐷 / 𝑑][𝐸 / 𝑒]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑)
5	1, 4	bitri 267	1 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑)

Syntax hints:   ↔ wb 198  [wsbc 3633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-v 3387  df-sbc 3634

This theorem is referenced by:  6rexfrabdioph  38149  7rexfrabdioph  38150