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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 4640 | . 2 ⊢ {𝐴, 𝐵} = {𝑦 ∣ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)} | |
2 | nfpr.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfeq2 2912 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
4 | nfpr.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfeq2 2912 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐵 |
6 | 3, 5 | nfor 1899 | . . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∨ 𝑦 = 𝐵) |
7 | 6 | nfab 2901 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)} |
8 | 1, 7 | nfcxfr 2893 | 1 ⊢ Ⅎ𝑥{𝐴, 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 844 = wceq 1533 {cab 2701 Ⅎwnfc 2875 {cpr 4623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-v 3468 df-un 3946 df-sn 4622 df-pr 4624 |
This theorem is referenced by: nfsn 4704 nfop 4882 nfaltop 35448 |
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