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

Step	Hyp	Ref	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)
3	2	sbcbii 3524	. . 3 ⊢ ([𝐶 / 𝑐][𝐴 / 𝑎][𝐷 / 𝑑][𝐸 / 𝑒]𝜑 ↔ [𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑)
4	3	sbcbii 3524	. 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 264	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 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