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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 4610 | . 2 ⊢ {𝐴, 𝐵} = {𝑦 ∣ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)} | |
2 | nfpr.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfeq2 2925 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
4 | nfpr.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfeq2 2925 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐵 |
6 | 3, 5 | nfor 1908 | . . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∨ 𝑦 = 𝐵) |
7 | 6 | nfab 2914 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)} |
8 | 1, 7 | nfcxfr 2906 | 1 ⊢ Ⅎ𝑥{𝐴, 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 846 = wceq 1542 {cab 2714 Ⅎwnfc 2888 {cpr 4593 |
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 2708 |
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 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-v 3450 df-un 3920 df-sn 4592 df-pr 4594 |
This theorem is referenced by: nfsn 4673 nfop 4851 nfaltop 34594 |
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