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Mirrors > Home > MPE Home > Th. List > cbvrexfw | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. Version of cbvrexf 3440 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by FL, 27-Apr-2008.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
cbvrexfw.1 | ⊢ Ⅎ𝑥𝐴 |
cbvrexfw.2 | ⊢ Ⅎ𝑦𝐴 |
cbvrexfw.3 | ⊢ Ⅎ𝑦𝜑 |
cbvrexfw.4 | ⊢ Ⅎ𝑥𝜓 |
cbvrexfw.5 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvrexfw | ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvrexfw.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | cbvrexfw.2 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
3 | cbvrexfw.3 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
4 | 3 | nfn 1857 | . . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑 |
5 | cbvrexfw.4 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
6 | 5 | nfn 1857 | . . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓 |
7 | cbvrexfw.5 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
8 | 7 | notbid 320 | . . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓)) |
9 | 1, 2, 4, 6, 8 | cbvralfw 3437 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜓) |
10 | 9 | notbii 322 | . 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓) |
11 | dfrex2 3239 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑) | |
12 | dfrex2 3239 | . 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 2961 ∀wral 3138 ∃wrex 3139 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 |
This theorem is referenced by: cbvrexw 3442 reusv2lem4 5302 reusv2 5304 nnwof 12315 cbviunf 30307 ac6sf2 30370 dfimafnf 30381 aciunf1lem 30407 bnj1400 32107 phpreu 34891 poimirlem26 34933 indexa 35023 evth2f 41321 fvelrnbf 41324 evthf 41333 eliin2f 41419 stoweidlem34 42368 ovnlerp 42893 |
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