![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > csbnestgw | Structured version Visualization version GIF version |
Description: Nest the composition of two substitutions. Version of csbnestg 4419 with a disjoint variable condition, which does not require ax-13 2363. (Contributed by NM, 23-Nov-2005.) Avoid ax-13 2363. (Revised by Gino Giotto, 26-Jan-2024.) |
Ref | Expression |
---|---|
csbnestgw | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2895 | . . 3 ⊢ Ⅎ𝑥𝐶 | |
2 | 1 | ax-gen 1789 | . 2 ⊢ ∀𝑦Ⅎ𝑥𝐶 |
3 | csbnestgfw 4412 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶) | |
4 | 2, 3 | mpan2 688 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 = wceq 1533 ∈ wcel 2098 Ⅎwnfc 2875 ⦋csb 3886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-v 3468 df-sbc 3771 df-csb 3887 |
This theorem is referenced by: disjxpin 32291 poimirlem24 37006 cdleme31snd 39751 cdlemeg46c 39878 |
Copyright terms: Public domain | W3C validator |