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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 1472 | . . 3 ⊢ (∀𝑥𝜑 → 𝜑) | |
2 | hb.2 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
3 | 1, 2 | imim12i 59 | . 2 ⊢ ((𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) |
4 | ax-i5r 1500 | . 2 ⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓)) | |
5 | hb.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
6 | 5 | imim1i 60 | . . 3 ⊢ ((∀𝑥𝜑 → 𝜓) → (𝜑 → 𝜓)) |
7 | 6 | alimi 1416 | . 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 1314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-5 1408 ax-gen 1410 ax-4 1472 ax-i5r 1500 |
This theorem is referenced by: hbbi 1512 hbia1 1516 19.21h 1521 19.38 1639 hbsbv 1894 hbmo1 2015 hbmo 2016 moexexdc 2061 2eu4 2070 cleqh 2217 hbral 2441 |
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