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Mirrors > Home > NFE Home > Th. List > nfral | GIF version |
Description: Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Ref | Expression |
---|---|
nfral.1 | ⊢ ℲxA |
nfral.2 | ⊢ Ⅎxφ |
Ref | Expression |
---|---|
nfral | ⊢ Ⅎx∀y ∈ A φ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1554 | . . 3 ⊢ Ⅎy ⊤ | |
2 | nfral.1 | . . . 4 ⊢ ℲxA | |
3 | 2 | a1i 10 | . . 3 ⊢ ( ⊤ → ℲxA) |
4 | nfral.2 | . . . 4 ⊢ Ⅎxφ | |
5 | 4 | a1i 10 | . . 3 ⊢ ( ⊤ → Ⅎxφ) |
6 | 1, 3, 5 | nfrald 2665 | . 2 ⊢ ( ⊤ → Ⅎx∀y ∈ A φ) |
7 | 6 | trud 1323 | 1 ⊢ Ⅎx∀y ∈ A φ |
Colors of variables: wff setvar class |
Syntax hints: ⊤ wtru 1316 Ⅎwnf 1544 Ⅎwnfc 2476 ∀wral 2614 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 |
This theorem is referenced by: nfra2 2668 nfrex 2669 rspc2 2960 sbcralt 3118 sbcralg 3120 raaan 3657 nfint 3936 nfiin 3996 ralxpf 4827 fun11iun 5305 dff13f 5472 nfiso 5487 mpt2eq123 5661 fmpt2x 5730 |
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