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Mirrors > Home > ILE Home > Th. List > nfralxy | GIF version |
Description: Not-free for restricted universal quantification where 𝑥 and 𝑦 are distinct. See nfralya 2450 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.) |
Ref | Expression |
---|---|
nfralxy.1 | ⊢ Ⅎ𝑥𝐴 |
nfralxy.2 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfralxy | ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1427 | . . 3 ⊢ Ⅎ𝑦⊤ | |
2 | nfralxy.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | a1i 9 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
4 | nfralxy.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | a1i 9 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
6 | 1, 3, 5 | nfraldxy 2444 | . 2 ⊢ (⊤ → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑) |
7 | 6 | mptru 1325 | 1 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff set class |
Syntax hints: ⊤wtru 1317 Ⅎwnf 1421 Ⅎwnfc 2245 ∀wral 2393 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-4 1472 ax-17 1491 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 |
This theorem is referenced by: nfra2xy 2452 rspc2 2774 sbcralt 2957 sbcralg 2959 raaanlem 3438 nfint 3751 nfiinxy 3810 nfpo 4193 nfso 4194 nfse 4233 nffrfor 4240 nfwe 4247 ralxpf 4655 funimaexglem 5176 fun11iun 5356 dff13f 5639 nfiso 5675 mpoeq123 5798 nfofr 5956 fmpox 6066 nfrecs 6172 xpf1o 6706 ac6sfi 6760 ismkvnex 6997 lble 8673 fzrevral 9853 nfsum1 11093 nfsum 11094 fsum2dlemstep 11171 fisumcom2 11175 bezoutlemmain 11613 cnmpt21 12387 setindis 13092 bdsetindis 13094 isomninnlem 13152 |
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