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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 4604 | . 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 1907 | . . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∨ 𝑦 = 𝐵) |
7 | 6 | nfab 2912 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)} |
8 | 1, 7 | nfcxfr 2904 | 1 ⊢ Ⅎ𝑥{𝐴, 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 845 = wceq 1541 {cab 2713 Ⅎwnfc 2886 {cpr 4587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-v 3446 df-un 3914 df-sn 4586 df-pr 4588 |
This theorem is referenced by: nfsn 4667 nfop 4845 nfaltop 34554 |
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