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Mirrors > Home > MPE Home > Th. List > nfim | Structured version Visualization version GIF version |
Description: If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 → 𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1701 changed. (Revised by Wolf Lammen, 17-Sep-2021.) |
Ref | Expression |
---|---|
nfim.1 | ⊢ Ⅎ𝑥𝜑 |
nfim.2 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
nfim | ⊢ Ⅎ𝑥(𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfim.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
3 | nfim.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜓) |
5 | 2, 4 | nfimd 1812 | . 2 ⊢ (⊤ → Ⅎ𝑥(𝜑 → 𝜓)) |
6 | 5 | trud 1484 | 1 ⊢ Ⅎ𝑥(𝜑 → 𝜓) |
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