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Mirrors > Home > MPE Home > Th. List > cbvrmow | Structured version Visualization version GIF version |
Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmo 3357 with a disjoint variable condition, which does not require ax-10 2141, ax-13 2371. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 23-May-2024.) |
Ref | Expression |
---|---|
cbvrmow.1 | ⊢ Ⅎ𝑦𝜑 |
cbvrmow.2 | ⊢ Ⅎ𝑥𝜓 |
cbvrmow.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvrmow | ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1922 | . . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
2 | cbvrmow.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | 1, 2 | nfan 1907 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) |
4 | nfv 1922 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 | |
5 | cbvrmow.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
6 | 4, 5 | nfan 1907 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓) |
7 | eleq1w 2820 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
8 | cbvrmow.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
9 | 7, 8 | anbi12d 634 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓))) |
10 | 3, 6, 9 | cbvmow 2602 | . 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
11 | df-rmo 3069 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
12 | df-rmo 3069 | . 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜓 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
13 | 10, 11, 12 | 3bitr4i 306 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 Ⅎwnf 1791 ∈ wcel 2110 ∃*wmo 2537 ∃*wrmo 3064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-mo 2539 df-clel 2816 df-rmo 3069 |
This theorem is referenced by: cbvdisj 5028 |
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