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Mirrors > Home > MPE Home > Th. List > cbvrex2v | Structured version Visualization version GIF version |
Description: Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.) |
Ref | Expression |
---|---|
cbvrex2v.1 | ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒)) |
cbvrex2v.2 | ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvrex2v | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvrex2v.1 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒)) | |
2 | 1 | rexbidv 3034 | . . 3 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜒)) |
3 | 2 | cbvrexv 3148 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) |
4 | cbvrex2v.2 | . . . 4 ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) | |
5 | 4 | cbvrexv 3148 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑤 ∈ 𝐵 𝜓) |
6 | 5 | rexbii 3023 | . 2 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) |
7 | 3, 6 | bitri 263 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∃wrex 2897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 |
This theorem is referenced by: omeu 7552 oeeui 7569 eroveu 7729 genpv 9700 bezoutlem3 15096 bezoutlem4 15097 bezout 15098 4sqlem2 15491 vdwnn 15540 efgrelexlema 17985 dyadmax 23172 2sqlem9 24952 2sq 24955 legov 25280 dfcgra2 25521 pstmfval 29267 nn0prpwlem 31487 isbnd2 32752 fourierdlem42 39042 fourierdlem54 39053 |
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