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Mirrors > Home > NFE Home > Th. List > nfexd | GIF version |
Description: If x is not free in φ, it is not free in ∃yφ. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Ref | Expression |
---|---|
nfald.1 | ⊢ Ⅎyφ |
nfald.2 | ⊢ (φ → Ⅎxψ) |
Ref | Expression |
---|---|
nfexd | ⊢ (φ → Ⅎx∃yψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ex 1542 | . 2 ⊢ (∃yψ ↔ ¬ ∀y ¬ ψ) | |
2 | nfald.1 | . . . 4 ⊢ Ⅎyφ | |
3 | nfald.2 | . . . . 5 ⊢ (φ → Ⅎxψ) | |
4 | 3 | nfnd 1791 | . . . 4 ⊢ (φ → Ⅎx ¬ ψ) |
5 | 2, 4 | nfald 1852 | . . 3 ⊢ (φ → Ⅎx∀y ¬ ψ) |
6 | 5 | nfnd 1791 | . 2 ⊢ (φ → Ⅎx ¬ ∀y ¬ ψ) |
7 | 1, 6 | nfxfrd 1571 | 1 ⊢ (φ → Ⅎx∃yψ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1540 ∃wex 1541 Ⅎwnf 1544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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 |
This theorem depends on definitions: df-bi 177 df-ex 1542 df-nf 1545 |
This theorem is referenced by: nfeld 2504 |
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