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Mirrors > Home > MPE Home > Th. List > nfint | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
nfint.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfint | ⊢ Ⅎ𝑥∩ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfint2 4945 | . 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧} | |
2 | nfint.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfv 1917 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧 | |
4 | 2, 3 | nfralw 3307 | . . 3 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧 |
5 | 4 | nfab 2908 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧} |
6 | 1, 5 | nfcxfr 2900 | 1 ⊢ Ⅎ𝑥∩ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: {cab 2708 Ⅎwnfc 2882 ∀wral 3060 ∩ cint 4943 |
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 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-int 4944 |
This theorem is referenced by: onminsb 7765 oawordeulem 8537 nnawordex 8620 rankidb 9777 cardmin2 9976 cardaleph 10066 cardmin 10541 sltval2 27086 ldsysgenld 32989 aomclem8 41574 naddwordnexlem4 41923 |
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