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Mirrors > Home > MPE Home > Th. List > cbvcsbv | Structured version Visualization version GIF version |
Description: Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
cbvcsbv.1 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbvcsbv | ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2917 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2917 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbvcsbv.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbvcsbw 3811 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ⦋csb 3801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-11 2159 ax-12 2176 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-ex 1783 df-nf 1787 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-sbc 3694 df-csb 3802 |
This theorem is referenced by: pmatcollpw3lem 21468 poimirlem27 35349 cdleme40v 38030 |
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