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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 2600 with a disjoint variable condition, which does not require ax-10 2138, ax-13 2372. (Contributed by NM, 9-Mar-1995.) (Revised by Gino Giotto, 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 1918 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 = 𝑧 | |
3 | 1, 2 | nfim 1900 | . . . 4 ⊢ Ⅎ𝑦(𝜑 → 𝑥 = 𝑧) |
4 | cbvmow.2 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
5 | nfv 1918 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 = 𝑧 | |
6 | 4, 5 | nfim 1900 | . . . 4 ⊢ Ⅎ𝑥(𝜓 → 𝑦 = 𝑧) |
7 | cbvmow.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
8 | equequ1 2029 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) | |
9 | 7, 8 | imbi12d 345 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜓 → 𝑦 = 𝑧))) |
10 | 3, 6, 9 | cbvalv1 2338 | . . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑦(𝜓 → 𝑦 = 𝑧)) |
11 | 10 | exbii 1851 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) |
12 | df-mo 2535 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)) | |
13 | df-mo 2535 | . 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 205 ∀wal 1540 ∃wex 1782 Ⅎwnf 1786 ∃*wmo 2533 |
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 1914 ax-6 1972 ax-7 2012 ax-11 2155 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-nf 1787 df-mo 2535 |
This theorem is referenced by: cbveuw 2602 cbvrmow 3406 dffun6f 6562 opabiotafun 6973 2ndcdisj 22960 cbvdisjf 31802 phpreu 36472 mo0sn 47500 isthincd2lem1 47647 |
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