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Mirrors > Home > MPE Home > Th. List > cbvexd | Structured version Visualization version GIF version |
Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2416. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Ref | Expression |
---|---|
cbvald.1 | ⊢ Ⅎ𝑦𝜑 |
cbvald.2 | ⊢ (𝜑 → Ⅎ𝑦𝜓) |
cbvald.3 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
cbvexd | ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvald.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | cbvald.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
3 | 2 | nfnd 1903 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦 ¬ 𝜓) |
4 | cbvald.3 | . . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
5 | notbi 311 | . . . . 5 ⊢ ((𝜓 ↔ 𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒)) | |
6 | 4, 5 | syl6ib 243 | . . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (¬ 𝜓 ↔ ¬ 𝜒))) |
7 | 1, 3, 6 | cbvald 2371 | . . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒)) |
8 | alnex 1825 | . . 3 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓) | |
9 | alnex 1825 | . . 3 ⊢ (∀𝑦 ¬ 𝜒 ↔ ¬ ∃𝑦𝜒) | |
10 | 7, 8, 9 | 3bitr3g 305 | . 2 ⊢ (𝜑 → (¬ ∃𝑥𝜓 ↔ ¬ ∃𝑦𝜒)) |
11 | 10 | con4bid 309 | 1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∀wal 1599 ∃wex 1823 Ⅎwnf 1827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-ex 1824 df-nf 1828 |
This theorem is referenced by: vtoclgft 3455 dfid3 5262 axrepndlem2 9750 axunnd 9753 axpowndlem2 9755 axpownd 9758 axregndlem2 9760 axinfndlem1 9762 axacndlem4 9767 wl-mo2df 33960 wl-eudf 33962 wl-mo2t 33965 |
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