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| Mirrors > Home > MPE Home > Th. List > sbcnestgw | Structured version Visualization version GIF version | ||
| Description: Nest the composition of two substitutions. Version of sbcnestg 4428 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 27-Nov-2005.) Avoid ax-13 2377. (Revised by GG, 26-Jan-2024.) |
| Ref | Expression |
|---|---|
| sbcnestgw | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfv 1914 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
| 2 | 1 | ax-gen 1795 | . 2 ⊢ ∀𝑦Ⅎ𝑥𝜑 |
| 3 | sbcnestgfw 4421 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) | |
| 4 | 2, 3 | mpan2 691 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 Ⅎwnf 1783 ∈ wcel 2108 [wsbc 3788 ⦋csb 3899 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-v 3482 df-sbc 3789 df-csb 3900 |
| This theorem is referenced by: sbcco3gw 4425 sbcop 5494 |
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