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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcrot5 | Structured version Visualization version GIF version |
Description: Rotate a sequence of five explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.) |
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) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcrot3 39732 | . 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 39732 | . . . 4 ⊢ ([𝐴 / 𝑎][𝐷 / 𝑑][𝐸 / 𝑒]𝜑 ↔ [𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑) | |
3 | 2 | sbcbii 3776 | . . 3 ⊢ ([𝐶 / 𝑐][𝐴 / 𝑎][𝐷 / 𝑑][𝐸 / 𝑒]𝜑 ↔ [𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑) |
4 | 3 | sbcbii 3776 | . 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 278 | 1 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 [wsbc 3720 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-sbc 3721 |
This theorem is referenced by: 6rexfrabdioph 39740 7rexfrabdioph 39741 |
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