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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 4421 with a disjoint variable condition, which does not require ax-13 2365. (Contributed by NM, 27-Nov-2005.) Avoid ax-13 2365. (Revised by Gino Giotto, 26-Jan-2024.) |
Ref | Expression |
---|---|
sbcnestgw | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1909 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | ax-gen 1789 | . 2 ⊢ ∀𝑦Ⅎ𝑥𝜑 |
3 | sbcnestgfw 4414 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) | |
4 | 2, 3 | mpan2 689 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1531 Ⅎwnf 1777 ∈ wcel 2098 [wsbc 3768 ⦋csb 3884 |
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 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-v 3465 df-sbc 3769 df-csb 3885 |
This theorem is referenced by: sbcco3gw 4418 sbcop 5485 |
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