Theorem sbcrot3 43991

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

Step	Hyp	Ref	Expression
1		sbccom 3822	. 2 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐]𝜑 ↔ [𝐵 / 𝑏][𝐴 / 𝑎][𝐶 / 𝑐]𝜑)
2		sbccom 3822	. . 3 ⊢ ([𝐴 / 𝑎][𝐶 / 𝑐]𝜑 ↔ [𝐶 / 𝑐][𝐴 / 𝑎]𝜑)
3	2	sbcbii 3798	. 2 ⊢ ([𝐵 / 𝑏][𝐴 / 𝑎][𝐶 / 𝑐]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎]𝜑)
4	1, 3	bitri 278	1 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎]𝜑)

This proof depends on syntax axioms:   ↔ wb 209  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-sbc 3743

This theorem is used by:  sbcrot5  43992  2rexfrabdioph  43996  3rexfrabdioph  43997  4rexfrabdioph  43998  7rexfrabdioph  44000