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Mirrors > Home > MPE Home > Th. List > cbvrexcsf | Structured version Visualization version GIF version |
Description: A more general version of cbvrexf 3442 that has no distinct variable restrictions. Changes bound variables using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2390. (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 1857 | . . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑 |
5 | cbvralcsf.4 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
6 | 5 | nfn 1857 | . . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓 |
7 | cbvralcsf.5 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
8 | cbvralcsf.6 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
9 | 8 | notbid 320 | . . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓)) |
10 | 1, 2, 4, 6, 7, 9 | cbvralcsf 3927 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝜓) |
11 | 10 | notbii 322 | . 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜓) |
12 | dfrex2 3241 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑) | |
13 | dfrex2 3241 | . 2 ⊢ (∃𝑦 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜓) | |
14 | 11, 12, 13 | 3bitr4i 305 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1537 Ⅎwnf 1784 Ⅎwnfc 2963 ∀wral 3140 ∃wrex 3141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-sbc 3775 df-csb 3886 |
This theorem is referenced by: cbvrexv2 3932 |
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