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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 4412 | . 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧} | |
2 | nfint.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧 | |
4 | 2, 3 | nfral 2929 | . . 3 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧 |
5 | 4 | nfab 2755 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧} |
6 | 1, 5 | nfcxfr 2749 | 1 ⊢ Ⅎ𝑥∩ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: {cab 2596 Ⅎwnfc 2738 ∀wral 2896 ∩ cint 4410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-int 4411 |
This theorem is referenced by: onminsb 6891 oawordeulem 7521 nnawordex 7604 rankidb 8546 cardmin2 8707 cardaleph 8795 cardmin 9265 ldsysgenld 29550 sltval2 31053 nobndlem5 31095 aomclem8 36649 |
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