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Mirrors > Home > MPE Home > Th. List > nfrabwOLD | Structured version Visualization version GIF version |
Description: Obsolete version of nfrabw 3439 as of 23-Nov2024. (Contributed by NM, 13-Oct-2003.) (Revised by Gino Giotto, 10-Jan-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
nfrabw.1 | ⊢ Ⅎ𝑥𝜑 |
nfrabw.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfrabwOLD | ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3407 | . 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} | |
2 | nftru 1807 | . . . 4 ⊢ Ⅎ𝑦⊤ | |
3 | nfrabw.2 | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | nfcri 2893 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
5 | 4 | a1i 11 | . . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑦 ∈ 𝐴) |
6 | nfrabw.1 | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
7 | 6 | a1i 11 | . . . . 5 ⊢ (⊤ → Ⅎ𝑥𝜑) |
8 | 5, 7 | nfand 1901 | . . . 4 ⊢ (⊤ → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) |
9 | 2, 8 | nfabdw 2929 | . . 3 ⊢ (⊤ → Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) |
10 | 9 | mptru 1549 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} |
11 | 1, 10 | nfcxfr 2904 | 1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ⊤wtru 1543 Ⅎwnf 1786 ∈ wcel 2107 {cab 2715 Ⅎwnfc 2886 {crab 3406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-rab 3407 |
This theorem is referenced by: (None) |
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