| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > cbvmow | Structured version Visualization version GIF version | ||
| Description: Rule used to change bound variables, using implicit substitution. Version of cbvmo 2604 with a disjoint variable condition, which does not require ax-10 2141, ax-13 2377. (Contributed by NM, 9-Mar-1995.) (Revised by GG, 23-May-2024.) |
| Ref | Expression |
|---|---|
| cbvmow.1 | ⊢ Ⅎ𝑦𝜑 |
| cbvmow.2 | ⊢ Ⅎ𝑥𝜓 |
| cbvmow.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| cbvmow | ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvmow.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
| 2 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 = 𝑧 | |
| 3 | 1, 2 | nfim 1896 | . . . 4 ⊢ Ⅎ𝑦(𝜑 → 𝑥 = 𝑧) |
| 4 | cbvmow.2 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
| 5 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 = 𝑧 | |
| 6 | 4, 5 | nfim 1896 | . . . 4 ⊢ Ⅎ𝑥(𝜓 → 𝑦 = 𝑧) |
| 7 | cbvmow.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 8 | equequ1 2024 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) | |
| 9 | 7, 8 | imbi12d 344 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜓 → 𝑦 = 𝑧))) |
| 10 | 3, 6, 9 | cbvalv1 2343 | . . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑦(𝜓 → 𝑦 = 𝑧)) |
| 11 | 10 | exbii 1848 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) |
| 12 | df-mo 2540 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)) | |
| 13 | df-mo 2540 | . 2 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) | |
| 14 | 11, 12, 13 | 3bitr4i 303 | 1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 ∃wex 1779 Ⅎwnf 1783 ∃*wmo 2538 |
| 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 1967 ax-7 2007 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-mo 2540 |
| This theorem is referenced by: cbveuw 2606 cbvrmow 3409 dffun6f 6579 opabiotafun 6989 2ndcdisj 23464 cbvdisjf 32584 phpreu 37611 mo0sn 48735 isthincd2lem1 49075 |
| Copyright terms: Public domain | W3C validator |