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Mirrors > Home > MPE Home > Th. List > nfcsbd | Structured version Visualization version GIF version |
Description: Deduction version of nfcsb 3835. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Ref | Expression |
---|---|
nfcsbd.1 | ⊢ Ⅎ𝑦𝜑 |
nfcsbd.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfcsbd.3 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
Ref | Expression |
---|---|
nfcsbd | ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-csb 3812 | . 2 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵} | |
2 | nfv 1892 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
3 | nfcsbd.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | nfcsbd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | nfcsbd.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
6 | 5 | nfcrd 2941 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐵) |
7 | 3, 4, 6 | nfsbcd 3727 | . . 3 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵) |
8 | 2, 7 | nfabd 2969 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}) |
9 | 1, 8 | nfcxfrd 2948 | 1 ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1765 ∈ wcel 2081 {cab 2775 Ⅎwnfc 2933 [wsbc 3706 ⦋csb 3811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-sbc 3707 df-csb 3812 |
This theorem is referenced by: nfcsb 3835 |
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