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| Mirrors > Home > MPE Home > Th. List > cbvrexcsf | Structured version Visualization version GIF version | ||
| Description: A more general version of cbvrexf 3360 that has no distinct variable restrictions. Changes bound variables using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2376. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| cbvralcsf.1 | ⊢ Ⅎ𝑦𝐴 | 
| cbvralcsf.2 | ⊢ Ⅎ𝑥𝐵 | 
| cbvralcsf.3 | ⊢ Ⅎ𝑦𝜑 | 
| cbvralcsf.4 | ⊢ Ⅎ𝑥𝜓 | 
| cbvralcsf.5 | ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | 
| cbvralcsf.6 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| cbvrexcsf | ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cbvralcsf.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
| 2 | cbvralcsf.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
| 3 | cbvralcsf.3 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
| 4 | 3 | nfn 1856 | . . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑 | 
| 5 | cbvralcsf.4 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
| 6 | 5 | nfn 1856 | . . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓 | 
| 7 | cbvralcsf.5 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
| 8 | cbvralcsf.6 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 9 | 8 | notbid 318 | . . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓)) | 
| 10 | 1, 2, 4, 6, 7, 9 | cbvralcsf 3940 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝜓) | 
| 11 | 10 | notbii 320 | . 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜓) | 
| 12 | dfrex2 3072 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑) | |
| 13 | dfrex2 3072 | . 2 ⊢ (∃𝑦 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜓) | |
| 14 | 11, 12, 13 | 3bitr4i 303 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1539 Ⅎwnf 1782 Ⅎwnfc 2889 ∀wral 3060 ∃wrex 3069 | 
| 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-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-13 2376 ax-ext 2707 | 
| 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-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-sbc 3788 df-csb 3899 | 
| This theorem is referenced by: cbvrexv2 3945 | 
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