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Mirrors > Home > ILE Home > Th. List > nfcsb1d | GIF version |
Description: Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.) |
Ref | Expression |
---|---|
nfcsb1d.1 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Ref | Expression |
---|---|
nfcsb1d | ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-csb 3050 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
2 | nfv 1521 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcsb1d.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
4 | 3 | nfsbc1d 2971 | . . 3 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝑦 ∈ 𝐵) |
5 | 2, 4 | nfabd 2332 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}) |
6 | 1, 5 | nfcxfrd 2310 | 1 ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2141 {cab 2156 Ⅎwnfc 2299 [wsbc 2955 ⦋csb 3049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-sbc 2956 df-csb 3050 |
This theorem is referenced by: nfcsb1 3081 |
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