| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > hbex | Structured version Visualization version GIF version | ||
| Description: If 𝑥 is not free in 𝜑, then it is not free in ∃𝑦𝜑. (Contributed by NM, 12-Mar-1993.) Reduce symbol count in nfex 2324, hbex 2325. (Revised by Wolf Lammen, 16-Oct-2021.) |
| Ref | Expression |
|---|---|
| hbex.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
| Ref | Expression |
|---|---|
| hbex | ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hbex.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
| 2 | 1 | nf5i 2146 | . . 3 ⊢ Ⅎ𝑥𝜑 |
| 3 | 2 | nfex 2324 | . 2 ⊢ Ⅎ𝑥∃𝑦𝜑 |
| 4 | 3 | nf5ri 2195 | 1 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 ∃wex 1779 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-or 849 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |