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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 3445 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-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 | cbvrmow.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | cbvrmow.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvrmow.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvrexw 3439 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) |
5 | 1, 2, 3 | cbvreuw 3440 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) |
6 | 4, 5 | imbi12i 353 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑦 ∈ 𝐴 𝜓 → ∃!𝑦 ∈ 𝐴 𝜓)) |
7 | rmo5 3431 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) | |
8 | rmo5 3431 | . 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜓 ↔ (∃𝑦 ∈ 𝐴 𝜓 → ∃!𝑦 ∈ 𝐴 𝜓)) | |
9 | 6, 7, 8 | 3bitr4i 305 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 Ⅎwnf 1783 ∃wrex 3138 ∃!wreu 3139 ∃*wrmo 3140 |
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 1969 ax-7 2014 ax-8 2115 ax-10 2144 ax-11 2160 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 |
This theorem is referenced by: cbvdisj 5034 |
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