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Mirrors > Home > ILE Home > Th. List > hbex | GIF version |
Description: If 𝑥 is not free in 𝜑, it is not free in ∃𝑦𝜑. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |
Ref | Expression |
---|---|
hbex.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
hbex | ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbe1 1384 | . . 3 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑) | |
2 | 1 | hbal 1366 | . 2 ⊢ (∀𝑥∃𝑦𝜑 → ∀𝑦∀𝑥∃𝑦𝜑) |
3 | hbex.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
4 | 19.8a 1482 | . . 3 ⊢ (𝜑 → ∃𝑦𝜑) | |
5 | 3, 4 | alrimih 1358 | . 2 ⊢ (𝜑 → ∀𝑥∃𝑦𝜑) |
6 | 2, 5 | exlimih 1484 | 1 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 ∃wex 1381 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: nfex 1528 excomim 1553 19.12 1555 cbvexh 1638 cbvexdh 1801 hbsbv 1817 hbeu1 1910 hbmo 1939 moexexdc 1984 |
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