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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cbv2hv | Structured version Visualization version GIF version |
Description: Version of cbv2h 2425 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-cbv2hv.1 | ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓)) |
bj-cbv2hv.2 | ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) |
bj-cbv2hv.3 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
bj-cbv2hv | ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-cbv2hv.1 | . . 3 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓)) | |
2 | bj-cbv2hv.2 | . . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) | |
3 | bj-cbv2hv.3 | . . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
4 | biimp 217 | . . . 4 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒)) | |
5 | 3, 4 | syl6 35 | . . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) |
6 | 1, 2, 5 | bj-cbv1hv 34122 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒)) |
7 | equcomi 2023 | . . . . 5 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) | |
8 | biimpr 222 | . . . . 5 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 → 𝜓)) | |
9 | 7, 3, 8 | syl56 36 | . . . 4 ⊢ (𝜑 → (𝑦 = 𝑥 → (𝜒 → 𝜓))) |
10 | 2, 1, 9 | bj-cbv1hv 34122 | . . 3 ⊢ (∀𝑦∀𝑥𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓)) |
11 | 10 | alcoms 2161 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓)) |
12 | 6, 11 | impbid 214 | 1 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-10 2144 ax-11 2160 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-nf 1784 |
This theorem is referenced by: bj-cbv2v 34124 |
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