![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > nfex | GIF version |
Description: If 𝑥 is not free in 𝜑, it is not free in ∃𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) |
Ref | Expression |
---|---|
nfex.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfex | ⊢ Ⅎ𝑥∃𝑦𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfex.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | nfri 1519 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | 2 | hbex 1636 | . 2 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) |
4 | 3 | nfi 1462 | 1 ⊢ Ⅎ𝑥∃𝑦𝜑 |
Colors of variables: wff set class |
Syntax hints: Ⅎwnf 1460 ∃wex 1492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 |
This theorem depends on definitions: df-bi 117 df-nf 1461 |
This theorem is referenced by: eeor 1695 cbvexv1 1752 cbvex2 1922 eean 1931 nfsbv 1947 nfeu1 2037 nfeuv 2044 nfel 2328 ceqsex2 2777 nfopab 4070 nfopab2 4072 cbvopab1 4075 cbvopab1s 4077 repizf2 4161 copsex2t 4244 copsex2g 4245 euotd 4253 onintrab2im 4516 mosubopt 4690 nfco 4791 dfdmf 4819 dfrnf 4867 nfdm 4870 fv3 5537 nfoprab2 5922 nfoprab3 5923 nfoprab 5924 cbvoprab1 5944 cbvoprab2 5945 cbvoprab3 5948 cnvoprab 6232 ac6sfi 6895 cc3 7264 nfsum1 11357 nfsum 11358 fsum2dlemstep 11435 nfcprod1 11555 nfcprod 11556 fprod2dlemstep 11623 |
Copyright terms: Public domain | W3C validator |