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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 40911. The equivalent of this theorem without the bound variable change is rexlimddv 3250. Version of rexlimddvcbv 40913 with a disjoint variable condition, which does not require ax-13 2379. (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 3397 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜃 ↔ ∃𝑦 ∈ 𝐴 𝜒) |
4 | rexlimddvcbvw.2 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) | |
5 | 4 | rexlimdvaa 3244 | . . 3 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝜒 → 𝜓)) |
6 | 3, 5 | syl5bi 245 | . 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 209 ∧ wa 399 ∈ wcel 2111 ∃wrex 3107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-clel 2870 df-ral 3111 df-rex 3112 |
This theorem is referenced by: mnuprdlem1 40980 mnuprdlem2 40981 |
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