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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cbv1hv | Structured version Visualization version GIF version |
Description: Version of cbv1h 2405 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-cbv1hv.1 | ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓)) |
bj-cbv1hv.2 | ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) |
bj-cbv1hv.3 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) |
Ref | Expression |
---|---|
bj-cbv1hv | ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2148 | . 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑 | |
2 | nfa2 2170 | . 2 ⊢ Ⅎ𝑦∀𝑥∀𝑦𝜑 | |
3 | 2sp 2179 | . . . 4 ⊢ (∀𝑥∀𝑦𝜑 → 𝜑) | |
4 | bj-cbv1hv.1 | . . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (∀𝑥∀𝑦𝜑 → (𝜓 → ∀𝑦𝜓)) |
6 | 2, 5 | nf5d 2281 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → Ⅎ𝑦𝜓) |
7 | bj-cbv1hv.2 | . . . 4 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) | |
8 | 3, 7 | syl 17 | . . 3 ⊢ (∀𝑥∀𝑦𝜑 → (𝜒 → ∀𝑥𝜒)) |
9 | 1, 8 | nf5d 2281 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → Ⅎ𝑥𝜒) |
10 | bj-cbv1hv.3 | . . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) | |
11 | 3, 10 | syl 17 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) |
12 | 1, 2, 6, 9, 11 | cbv1v 2333 | 1 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-or 845 df-ex 1783 df-nf 1787 |
This theorem is referenced by: bj-cbv2hv 34979 |
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