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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 3366 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by FL, 2-Jul-2012.) Avoid ax-13 2372. (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 3179 | . . 3 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜒)) |
3 | 2 | cbvrexvw 3236 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) |
4 | cbvrex2vw.2 | . . . 4 ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) | |
5 | 4 | cbvrexvw 3236 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑤 ∈ 𝐵 𝜓) |
6 | 5 | rexbii 3095 | . 2 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) |
7 | 3, 6 | bitri 275 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∃wrex 3071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-clel 2811 df-rex 3072 |
This theorem is referenced by: omeu 8581 oeeui 8598 eroveu 8802 genpv 10990 bezoutlem3 16479 bezoutlem4 16480 bezout 16481 4sqlem2 16878 vdwnn 16927 efgrelexlema 19610 dyadmax 25097 2sqlem9 26910 2sq 26913 mulsval2lem 27546 precsexlemcbv 27632 legov 27816 dfcgra2 28061 pstmfval 32814 satfv0 34287 satfv0fun 34300 fmla1 34316 nn0prpwlem 35145 isbnd2 36589 nna4b4nsq 41346 oaun3lem1 42057 limsupref 44336 fourierdlem42 44800 fourierdlem54 44811 mogoldbb 46388 |
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