Theorem sbcrot3 43251

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 3805	. 2 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐]𝜑 ↔ [𝐵 / 𝑏][𝐴 / 𝑎][𝐶 / 𝑐]𝜑)
2		sbccom 3805	. . 3 ⊢ ([𝐴 / 𝑎][𝐶 / 𝑐]𝜑 ↔ [𝐶 / 𝑐][𝐴 / 𝑎]𝜑)
3	2	sbcbii 3781	. 2 ⊢ ([𝐵 / 𝑏][𝐴 / 𝑎][𝐶 / 𝑐]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎]𝜑)
4	1, 3	bitri 277	1 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎]𝜑)

Syntax hints:   ↔ wb 208  [wsbc 3725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-11 2170  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-sbc 3726

This theorem is referenced by:  sbcrot5  43252  2rexfrabdioph  43256  3rexfrabdioph  43257  4rexfrabdioph  43258  7rexfrabdioph  43260