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| Mirrors > Home > MPE Home > Th. List > nfpr | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.) |
| Ref | Expression |
|---|---|
| nfpr.1 | ⊢ Ⅎ𝑥𝐴 |
| nfpr.2 | ⊢ Ⅎ𝑥𝐵 |
| Ref | Expression |
|---|---|
| nfpr | ⊢ Ⅎ𝑥{𝐴, 𝐵} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfpr2 4646 | . 2 ⊢ {𝐴, 𝐵} = {𝑦 ∣ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)} | |
| 2 | nfpr.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
| 3 | 2 | nfeq2 2923 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
| 4 | nfpr.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
| 5 | 4 | nfeq2 2923 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐵 |
| 6 | 3, 5 | nfor 1904 | . . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∨ 𝑦 = 𝐵) |
| 7 | 6 | nfab 2911 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)} |
| 8 | 1, 7 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥{𝐴, 𝐵} |
| Colors of variables: wff setvar class |
| Syntax hints: ∨ wo 848 = wceq 1540 {cab 2714 Ⅎwnfc 2890 {cpr 4628 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: nfsn 4707 nfop 4889 nfaltop 35981 |
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