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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cbv1v | Structured version Visualization version GIF version |
Description: Version of cbv1 2255 with a dv condition, which does not require ax-13 2234. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-cbv1v.1 | ⊢ Ⅎ𝑥𝜑 |
bj-cbv1v.2 | ⊢ Ⅎ𝑦𝜑 |
bj-cbv1v.3 | ⊢ (𝜑 → Ⅎ𝑦𝜓) |
bj-cbv1v.4 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
bj-cbv1v.5 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) |
Ref | Expression |
---|---|
bj-cbv1v | ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-cbv1v.2 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
2 | bj-cbv1v.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
3 | 1, 2 | nfim1 2055 | . . . 4 ⊢ Ⅎ𝑦(𝜑 → 𝜓) |
4 | bj-cbv1v.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
5 | bj-cbv1v.4 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
6 | 4, 5 | nfim1 2055 | . . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜒) |
7 | bj-cbv1v.5 | . . . . . 6 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) | |
8 | 7 | com12 32 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝜓 → 𝜒))) |
9 | 8 | a2d 29 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) |
10 | 3, 6, 9 | cbv3v 2158 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑦(𝜑 → 𝜒)) |
11 | 4 | 19.21 2062 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) |
12 | 1 | 19.21 2062 | . . 3 ⊢ (∀𝑦(𝜑 → 𝜒) ↔ (𝜑 → ∀𝑦𝜒)) |
13 | 10, 11, 12 | 3imtr3i 279 | . 2 ⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑦𝜒)) |
14 | 13 | pm2.86i 107 | 1 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-ex 1696 df-nf 1701 |
This theorem is referenced by: bj-cbv1hv 31917 |
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