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Mirrors > Home > MPE Home > Th. List > nfcsb1d | Structured version Visualization version 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 3534 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
2 | nfv 1843 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcsb1d.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
4 | 3 | nfsbc1d 3453 | . . 3 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝑦 ∈ 𝐵) |
5 | 2, 4 | nfabd 2785 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}) |
6 | 1, 5 | nfcxfrd 2763 | 1 ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 {cab 2608 Ⅎwnfc 2751 [wsbc 3435 ⦋csb 3533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-sbc 3436 df-csb 3534 |
This theorem is referenced by: nfcsb1 3548 riotaeqimp 6634 |
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