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| Mirrors > Home > MPE Home > Th. List > nfcsbw | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3926 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Mario Carneiro, 12-Oct-2016.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) |
| Ref | Expression |
|---|---|
| nfcsbw.1 | ⊢ Ⅎ𝑥𝐴 |
| nfcsbw.2 | ⊢ Ⅎ𝑥𝐵 |
| Ref | Expression |
|---|---|
| nfcsbw | ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-csb 3900 | . . 3 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵} | |
| 2 | nftru 1804 | . . . 4 ⊢ Ⅎ𝑧⊤ | |
| 3 | nftru 1804 | . . . . 5 ⊢ Ⅎ𝑦⊤ | |
| 4 | nfcsbw.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
| 5 | 4 | a1i 11 | . . . . 5 ⊢ (⊤ → Ⅎ𝑥𝐴) |
| 6 | nfcsbw.2 | . . . . . . 7 ⊢ Ⅎ𝑥𝐵 | |
| 7 | 6 | a1i 11 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑥𝐵) |
| 8 | 7 | nfcrd 2899 | . . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑧 ∈ 𝐵) |
| 9 | 3, 5, 8 | nfsbcdw 3809 | . . . 4 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵) |
| 10 | 2, 9 | nfabdw 2927 | . . 3 ⊢ (⊤ → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}) |
| 11 | 1, 10 | nfcxfrd 2904 | . 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵) |
| 12 | 11 | mptru 1547 | 1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: ⊤wtru 1541 ∈ wcel 2108 {cab 2714 Ⅎwnfc 2890 [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-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-sbc 3789 df-csb 3900 |
| This theorem is referenced by: cbvrabcsfw 3940 elfvmptrab1w 7043 fmptcof 7150 fvmpopr2d 7595 elovmporab1w 7680 mpomptsx 8089 dmmpossx 8091 fmpox 8092 el2mpocsbcl 8110 fmpoco 8120 dfmpo 8127 mpocurryd 8294 fvmpocurryd 8296 nfsum 15727 fsum2dlem 15806 fsumcom2 15810 nfcprod 15945 fprod2dlem 16016 fprodcom2 16020 fsumcn 24894 fsum2cn 24895 dvmptfsum 26013 itgsubst 26090 iundisj2f 32603 f1od2 32732 esumiun 34095 poimirlem26 37653 cdlemkid 40938 cdlemk19x 40945 cdlemk11t 40948 fmpocos 42275 wdom2d2 43047 dmmpossx2 48253 |
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