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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rexcom4b | Structured version Visualization version GIF version |
Description: Remove from rexcom4b 3524 dependency on ax-ext 2793 and ax-13 2386 (and on df-or 844, df-cleq 2814, df-nfc 2963, df-v 3496). The hypothesis uses 𝑉 instead of V (see bj-isseti 34189 for the motivation). Use bj-rexcom4bv 34193 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-rexcom4b.1 | ⊢ 𝐵 ∈ 𝑉 |
Ref | Expression |
---|---|
bj-rexcom4b | ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom4a 3251 | . 2 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) | |
2 | bj-rexcom4b.1 | . . . . 5 ⊢ 𝐵 ∈ 𝑉 | |
3 | 2 | bj-isseti 34189 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝐵 |
4 | 3 | biantru 532 | . . 3 ⊢ (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) |
5 | 4 | rexbii 3247 | . 2 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) |
6 | 1, 5 | bitr4i 280 | 1 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ∃wrex 3139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-11 2157 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-sb 2066 df-clab 2800 df-clel 2893 df-rex 3144 |
This theorem is referenced by: (None) |
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