Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2469. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker cbvexdw 2355 if possible. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.) |
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 1854 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦 ¬ 𝜓) |
4 | cbvald.3 | . . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
5 | notbi 321 | . . . . 5 ⊢ ((𝜓 ↔ 𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒)) | |
6 | 4, 5 | syl6ib 253 | . . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (¬ 𝜓 ↔ ¬ 𝜒))) |
7 | 1, 3, 6 | cbvald 2424 | . . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒)) |
8 | alnex 1778 | . . 3 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓) | |
9 | alnex 1778 | . . 3 ⊢ (∀𝑦 ¬ 𝜒 ↔ ¬ ∃𝑦𝜒) | |
10 | 7, 8, 9 | 3bitr3g 315 | . 2 ⊢ (𝜑 → (¬ ∃𝑥𝜓 ↔ ¬ ∃𝑦𝜒)) |
11 | 10 | con4bid 319 | 1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∀wal 1531 ∃wex 1776 Ⅎwnf 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-11 2157 ax-12 2173 ax-13 2386 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 |
This theorem is referenced by: cbvexdva 2427 vtoclgftOLD 3554 dfid3 5461 axrepndlem2 10014 axunnd 10017 axpowndlem2 10019 axpownd 10022 axregndlem2 10024 axinfndlem1 10026 axacndlem4 10031 wl-mo2df 34805 wl-eudf 34807 |
Copyright terms: Public domain | W3C validator |