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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rexcom4bv | Structured version Visualization version GIF version |
Description: Version of bj-rexcom4b 32872 with a dv condition on 𝑥, 𝑉, hence removing dependency on df-sb 1881 and df-clab 2609 (so that it depends on df-clel 2618 and df-rex 2918 only on top of first-order logic). Prefer its use over bj-rexcom4b 32872 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-rexcom4bv.1 | ⊢ 𝐵 ∈ 𝑉 |
Ref | Expression |
---|---|
bj-rexcom4bv | ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-rexcom4a 32870 | . 2 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) | |
2 | bj-rexcom4bv.1 | . . . . 5 ⊢ 𝐵 ∈ 𝑉 | |
3 | 2 | bj-issetiv 32863 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝐵 |
4 | 3 | biantru 526 | . . 3 ⊢ (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) |
5 | 4 | rexbii 3041 | . 2 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) |
6 | 1, 5 | bitr4i 267 | 1 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∃wrex 2913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-11 2034 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-clel 2618 df-rex 2918 |
This theorem is referenced by: (None) |
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