Theorem sbcrot3 43148

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 3823	. 2 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐]𝜑 ↔ [𝐵 / 𝑏][𝐴 / 𝑎][𝐶 / 𝑐]𝜑)
2		sbccom 3823	. . 3 ⊢ ([𝐴 / 𝑎][𝐶 / 𝑐]𝜑 ↔ [𝐶 / 𝑐][𝐴 / 𝑎]𝜑)
3	2	sbcbii 3799	. 2 ⊢ ([𝐵 / 𝑏][𝐴 / 𝑎][𝐶 / 𝑐]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎]𝜑)
4	1, 3	bitri 275	1 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎]𝜑)

Syntax hints:   ↔ wb 206  [wsbc 3742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-sbc 3743

This theorem is referenced by:  sbcrot5  43149  2rexfrabdioph  43153  3rexfrabdioph  43154  4rexfrabdioph  43155  7rexfrabdioph  43157