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Mirrors > Home > MPE Home > Th. List > cbvrex2vw | Structured version Visualization version GIF version |
Description: Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v 3465 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by FL, 2-Jul-2012.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
cbvrex2vw.1 | ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒)) |
cbvrex2vw.2 | ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvrex2vw | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvrex2vw.1 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒)) | |
2 | 1 | rexbidv 3297 | . . 3 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜒)) |
3 | 2 | cbvrexvw 3450 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) |
4 | cbvrex2vw.2 | . . . 4 ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) | |
5 | 4 | cbvrexvw 3450 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑤 ∈ 𝐵 𝜓) |
6 | 5 | rexbii 3247 | . 2 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) |
7 | 3, 6 | bitri 277 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∃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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-clel 2893 df-rex 3144 |
This theorem is referenced by: omeu 8211 oeeui 8228 eroveu 8392 genpv 10421 bezoutlem3 15889 bezoutlem4 15890 bezout 15891 4sqlem2 16285 vdwnn 16334 efgrelexlema 18875 dyadmax 24199 2sqlem9 26003 2sq 26006 legov 26371 dfcgra2 26616 pstmfval 31136 satfv0 32605 satfv0fun 32618 fmla1 32634 nn0prpwlem 33670 isbnd2 35076 limsupref 42015 fourierdlem42 42483 fourierdlem54 42494 mogoldbb 43999 |
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