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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 4370 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 27-Nov-2005.) (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 1914 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | ax-gen 1795 | . 2 ⊢ ∀𝑦Ⅎ𝑥𝜑 |
3 | sbcnestgfw 4363 | . 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 208 ∀wal 1534 Ⅎwnf 1783 ∈ wcel 2113 [wsbc 3768 ⦋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: sbcco3gw 4367 sbcop 5373 |
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