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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rexvw | Structured version Visualization version GIF version |
Description: A weak version of rexv 3519 not using ax-ext 2792 (nor df-cleq 2813, df-clel 2892, df-v 3495), and only core FOL axioms. See also bj-ralvw 34222. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-rexvw.1 | ⊢ 𝜓 |
Ref | Expression |
---|---|
bj-rexvw | ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 3143 | . 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)) | |
2 | bj-rexvw.1 | . . . . 5 ⊢ 𝜓 | |
3 | 2 | vexw 2804 | . . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓} |
4 | 3 | biantrur 533 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)) |
5 | 4 | exbii 1847 | . 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)) |
6 | 1, 5 | bitr4i 280 | 1 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∃wex 1779 ∈ wcel 2113 {cab 2798 ∃wrex 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-sb 2069 df-clab 2799 df-rex 3143 |
This theorem is referenced by: (None) |
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