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Mirrors > Home > NFE Home > Th. List > nfand | GIF version |
Description: If in a context x is not free in ψ and χ, it is not free in (ψ ∧ χ). (Contributed by Mario Carneiro, 7-Oct-2016.) |
Ref | Expression |
---|---|
nfand.1 | ⊢ (φ → Ⅎxψ) |
nfand.2 | ⊢ (φ → Ⅎxχ) |
Ref | Expression |
---|---|
nfand | ⊢ (φ → Ⅎx(ψ ∧ χ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-an 360 | . 2 ⊢ ((ψ ∧ χ) ↔ ¬ (ψ → ¬ χ)) | |
2 | nfand.1 | . . . 4 ⊢ (φ → Ⅎxψ) | |
3 | nfand.2 | . . . . 5 ⊢ (φ → Ⅎxχ) | |
4 | 3 | nfnd 1791 | . . . 4 ⊢ (φ → Ⅎx ¬ χ) |
5 | 2, 4 | nfimd 1808 | . . 3 ⊢ (φ → Ⅎx(ψ → ¬ χ)) |
6 | 5 | nfnd 1791 | . 2 ⊢ (φ → Ⅎx ¬ (ψ → ¬ χ)) |
7 | 1, 6 | nfxfrd 1571 | 1 ⊢ (φ → Ⅎx(ψ ∧ χ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 358 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-11 1746 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-nf 1545 |
This theorem is referenced by: nf3and 1823 nfan 1824 nfbid 1832 nfeld 2504 nfreud 2783 nfrmod 2784 nfrmo 2786 nfrab 2792 nfifd 3685 dfid3 4768 |
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