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Mirrors > Home > ILE Home > Th. List > nfrimi | GIF version |
Description: Moving an antecedent outside Ⅎ. (Contributed by Jim Kingdon, 23-Mar-2018.) |
Ref | Expression |
---|---|
nfrimi.1 | ⊢ Ⅎ𝑥𝜑 |
nfrimi.2 | ⊢ Ⅎ𝑥(𝜑 → 𝜓) |
Ref | Expression |
---|---|
nfrimi | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfrimi.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfrimi.2 | . . . . 5 ⊢ Ⅎ𝑥(𝜑 → 𝜓) | |
3 | 2 | nfri 1412 | . . . 4 ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) |
4 | 1 | nfri 1412 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) |
5 | ax-5 1336 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) | |
6 | 3, 4, 5 | syl2im 34 | . . 3 ⊢ ((𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)) |
7 | 6 | pm2.86i 92 | . 2 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
8 | 1, 7 | nfd 1416 | 1 ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 Ⅎwnf 1349 |
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-gen 1338 ax-4 1400 |
This theorem depends on definitions: df-bi 110 df-nf 1350 |
This theorem is referenced by: hbsbd 1858 |
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