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Mirrors > Home > MPE Home > Th. List > csbnestgw | Structured version Visualization version GIF version |
Description: Nest the composition of two substitutions. Version of csbnestg 4371 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 23-Nov-2005.) (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 2976 | . . 3 ⊢ Ⅎ𝑥𝐶 | |
2 | 1 | ax-gen 1795 | . 2 ⊢ ∀𝑦Ⅎ𝑥𝐶 |
3 | csbnestgfw 4364 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶) | |
4 | 2, 3 | mpan2 689 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1534 = wceq 1536 ∈ wcel 2113 Ⅎwnfc 2960 ⦋csb 3876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-v 3493 df-sbc 3769 df-csb 3877 |
This theorem is referenced by: disjxpin 30336 poimirlem24 34958 cdleme31snd 37562 cdlemeg46c 37689 |
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