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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 3364 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by FL, 2-Jul-2012.) Avoid ax-13 2370. (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 3177 | . . 3 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜒)) |
3 | 2 | cbvrexvw 3234 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) |
4 | cbvrex2vw.2 | . . . 4 ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) | |
5 | 4 | cbvrexvw 3234 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑤 ∈ 𝐵 𝜓) |
6 | 5 | rexbii 3093 | . 2 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) |
7 | 3, 6 | bitri 274 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∃wrex 3069 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1781 df-clel 2809 df-rex 3070 |
This theorem is referenced by: omeu 8588 oeeui 8605 eroveu 8809 genpv 10997 bezoutlem3 16488 bezoutlem4 16489 bezout 16490 4sqlem2 16887 vdwnn 16936 efgrelexlema 19659 dyadmax 25348 2sqlem9 27163 2sq 27166 mulsval2lem 27802 precsexlemcbv 27888 legov 28100 dfcgra2 28345 pstmfval 33171 satfv0 34644 satfv0fun 34657 fmla1 34673 nn0prpwlem 35511 isbnd2 36955 nna4b4nsq 41705 oaun3lem1 42427 limsupref 44701 fourierdlem42 45165 fourierdlem54 45176 mogoldbb 46753 |
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