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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 4386 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 27-Nov-2005.) Avoid ax-13 2371. (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 1918 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | ax-gen 1798 | . 2 ⊢ ∀𝑦Ⅎ𝑥𝜑 |
3 | sbcnestgfw 4379 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) | |
4 | 2, 3 | mpan2 690 | 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 1540 Ⅎwnf 1786 ∈ wcel 2107 [wsbc 3740 ⦋csb 3856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-v 3446 df-sbc 3741 df-csb 3857 |
This theorem is referenced by: sbcco3gw 4383 sbcop 5447 |
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