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| Mirrors > Home > MPE Home > Th. List > cbvrmo | Structured version Visualization version GIF version | ||
| Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker cbvrmow 3408, cbvrmovw 3402 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| cbvrmo.1 | ⊢ Ⅎ𝑦𝜑 | 
| cbvrmo.2 | ⊢ Ⅎ𝑥𝜓 | 
| cbvrmo.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| cbvrmo | ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cbvrmo.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 2 | cbvrmo.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
| 3 | cbvrmo.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 4 | 1, 2, 3 | cbvrex 3362 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) | 
| 5 | 1, 2, 3 | cbvreu 3427 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) | 
| 6 | 4, 5 | imbi12i 350 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑦 ∈ 𝐴 𝜓 → ∃!𝑦 ∈ 𝐴 𝜓)) | 
| 7 | rmo5 3399 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) | |
| 8 | rmo5 3399 | . 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜓 ↔ (∃𝑦 ∈ 𝐴 𝜓 → ∃!𝑦 ∈ 𝐴 𝜓)) | |
| 9 | 6, 7, 8 | 3bitr4i 303 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 Ⅎwnf 1782 ∃wrex 3069 ∃!wreu 3377 ∃*wrmo 3378 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-10 2140 ax-11 2156 ax-12 2176 ax-13 2376 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 | 
| This theorem is referenced by: cbvrmov 3429 | 
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