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Mirrors > Home > MPE Home > Th. List > cbvrexf | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker cbvrexfw 3440 when possible. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvralf.1 | ⊢ Ⅎ𝑥𝐴 |
cbvralf.2 | ⊢ Ⅎ𝑦𝐴 |
cbvralf.3 | ⊢ Ⅎ𝑦𝜑 |
cbvralf.4 | ⊢ Ⅎ𝑥𝜓 |
cbvralf.5 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvrexf | ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvralf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | cbvralf.2 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
3 | cbvralf.3 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
4 | 3 | nfn 1857 | . . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑 |
5 | cbvralf.4 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
6 | 5 | nfn 1857 | . . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓 |
7 | cbvralf.5 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
8 | 7 | notbid 320 | . . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓)) |
9 | 1, 2, 4, 6, 8 | cbvralf 3441 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜓) |
10 | 9 | notbii 322 | . 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓) |
11 | dfrex2 3241 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑) | |
12 | dfrex2 3241 | . 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓) | |
13 | 10, 11, 12 | 3bitr4i 305 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 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-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 |
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-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 |
This theorem is referenced by: cbvrex 3448 |
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