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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rexlimddvcbvw | Structured version Visualization version GIF version |
Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 42570. The equivalent of this theorem without the bound variable change is rexlimddv 3155. Version of rexlimddvcbv 42572 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Revised by Gino Giotto, 2-Apr-2024.) |
Ref | Expression |
---|---|
rexlimddvcbvw.1 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃) |
rexlimddvcbvw.2 | ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) |
rexlimddvcbvw.3 | ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexlimddvcbvw | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimddvcbvw.1 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃) | |
2 | rexlimddvcbvw.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒)) | |
3 | 2 | cbvrexvw 3225 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜃 ↔ ∃𝑦 ∈ 𝐴 𝜒) |
4 | rexlimddvcbvw.2 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) | |
5 | 4 | rexlimdvaa 3150 | . . 3 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝜒 → 𝜓)) |
6 | 3, 5 | biimtrid 241 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜃 → 𝜓)) |
7 | 1, 6 | mpd 15 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ∃wrex 3070 |
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 3071 |
This theorem is referenced by: mnuprdlem1 42644 mnuprdlem2 42645 |
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