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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rexcom4bv | Structured version Visualization version GIF version |
Description: Version of rexcom4b 3471 and bj-rexcom4b 34323 with a disjoint variable condition on 𝑥, 𝑉, hence removing dependency on df-sb 2070 and df-clab 2777 (so that it depends on df-clel 2870 and df-rex 3112 only on top of first-order logic). Prefer its use over bj-rexcom4b 34323 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 3214 | . 2 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) | |
2 | bj-rexcom4bv.1 | . . . . 5 ⊢ 𝐵 ∈ 𝑉 | |
3 | 2 | bj-issetiv 34317 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝐵 |
4 | 3 | biantru 533 | . . 3 ⊢ (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) |
5 | 4 | rexbii 3210 | . 2 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) |
6 | 1, 5 | bitr4i 281 | 1 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ 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 ax-11 2158 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-clel 2870 df-rex 3112 |
This theorem is referenced by: (None) |
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