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Mirrors > Home > ILE Home > Th. List > hbim | GIF version |
Description: If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 → 𝜓). (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (Revised by Mario Carneiro, 2-Feb-2015.) |
Ref | Expression |
---|---|
hb.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
hb.2 | ⊢ (𝜓 → ∀𝑥𝜓) |
Ref | Expression |
---|---|
hbim | ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-4 1400 | . . 3 ⊢ (∀𝑥𝜑 → 𝜑) | |
2 | hb.2 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
3 | 1, 2 | imim12i 53 | . 2 ⊢ ((𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) |
4 | ax-i5r 1428 | . 2 ⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓)) | |
5 | hb.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
6 | 5 | imim1i 54 | . . 3 ⊢ ((∀𝑥𝜑 → 𝜓) → (𝜑 → 𝜓)) |
7 | 6 | alimi 1344 | . 2 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) |
8 | 3, 4, 7 | 3syl 17 | 1 ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-5 1336 ax-gen 1338 ax-4 1400 ax-i5r 1428 |
This theorem is referenced by: hbbi 1440 hbia1 1444 19.21h 1449 19.38 1566 hbsbv 1817 hbmo1 1938 hbmo 1939 moexexdc 1984 2eu4 1993 cleqh 2137 hbral 2353 |
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