Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3887 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
2 | nfv 1914 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcsb1d.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
4 | 3 | nfsbc1d 3793 | . . 3 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝑦 ∈ 𝐵) |
5 | 2, 4 | nfabdw 3003 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}) |
6 | 1, 5 | nfcxfrd 2979 | 1 ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 {cab 2802 Ⅎwnfc 2964 [wsbc 3775 ⦋csb 3886 |
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 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-sbc 3776 df-csb 3887 |
This theorem is referenced by: nfcsb1 3909 riotaeqimp 7143 |
Copyright terms: Public domain | W3C validator |