Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
||
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 37355 | . 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 37355 | . . . 4 ⊢ ([𝐴 / 𝑎][𝐷 / 𝑑][𝐸 / 𝑒]𝜑 ↔ [𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑) | |
3 | 2 | sbcbii 3491 | . . 3 ⊢ ([𝐶 / 𝑐][𝐴 / 𝑎][𝐷 / 𝑑][𝐸 / 𝑒]𝜑 ↔ [𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑) |
4 | 3 | sbcbii 3491 | . 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 264 | 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 196 [wsbc 3435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-v 3202 df-sbc 3436 |
This theorem is referenced by: 6rexfrabdioph 37363 7rexfrabdioph 37364 |
Copyright terms: Public domain | W3C validator |