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Mirrors > Home > MPE Home > Th. List > csbnestg | Structured version Visualization version GIF version |
Description: Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker csbnestgw 4421 when possible. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
csbnestg | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2902 | . . 3 ⊢ Ⅎ𝑥𝐶 | |
2 | 1 | ax-gen 1796 | . 2 ⊢ ∀𝑦Ⅎ𝑥𝐶 |
3 | csbnestgf 4424 | . 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 1538 = wceq 1540 ∈ wcel 2105 Ⅎwnfc 2882 ⦋csb 3893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-13 2370 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-v 3475 df-sbc 3778 df-csb 3894 |
This theorem is referenced by: csbco3g 4428 |
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