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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcrot3 | Structured version Visualization version GIF version |
Description: Rotate a sequence of three explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
sbcrot3 | ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbccom 3861 | . 2 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐]𝜑 ↔ [𝐵 / 𝑏][𝐴 / 𝑎][𝐶 / 𝑐]𝜑) | |
2 | sbccom 3861 | . . 3 ⊢ ([𝐴 / 𝑎][𝐶 / 𝑐]𝜑 ↔ [𝐶 / 𝑐][𝐴 / 𝑎]𝜑) | |
3 | 2 | sbcbii 3833 | . 2 ⊢ ([𝐵 / 𝑏][𝐴 / 𝑎][𝐶 / 𝑐]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎]𝜑) |
4 | 1, 3 | bitri 274 | 1 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 [wsbc 3773 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-sbc 3774 |
This theorem is referenced by: sbcrot5 41301 2rexfrabdioph 41305 3rexfrabdioph 41306 4rexfrabdioph 41307 7rexfrabdioph 41309 |
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