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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rexcom4bv | Structured version Visualization version GIF version |
Description: Version of rexcom4b 3502 and bj-rexcom4b 36066 with a disjoint variable condition on 𝑥, 𝑉, hence removing dependency on df-sb 2066 and df-clab 2708 (so that it depends on df-clel 2808 and df-rex 3069 only on top of first-order logic). Prefer its use over bj-rexcom4b 36066 when sufficient (in particular when 𝑉 is substituted for V). Note the 𝑉 in the hypothesis instead of 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 | rexcom4a 3287 | . 2 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) | |
2 | bj-rexcom4bv.1 | . . . . 5 ⊢ 𝐵 ∈ 𝑉 | |
3 | 2 | bj-issetiv 36060 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝐵 |
4 | 3 | biantru 528 | . . 3 ⊢ (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) |
5 | 4 | rexbii 3092 | . 2 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) |
6 | 1, 5 | bitr4i 277 | 1 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1539 ∃wex 1779 ∈ wcel 2104 ∃wrex 3068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-11 2152 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1780 df-clel 2808 df-rex 3069 |
This theorem is referenced by: (None) |
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