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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 2899 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2899 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbvcsbv.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbvcsbw 3800 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ⦋csb 3790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-11 2162 ax-12 2179 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-sbc 3681 df-csb 3791 |
This theorem is referenced by: pmatcollpw3lem 21536 poimirlem27 35449 cdleme40v 38128 |
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