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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ralvw | Structured version Visualization version GIF version |
Description: A weak version of ralv 3516 not using ax-ext 2792 (nor df-cleq 2813, df-clel 2892, df-v 3493), and only core FOL axioms. See also bj-rexvw 34220. The analogues for reuv 3518 and rmov 3519 are not proved. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ralvw.1 | ⊢ 𝜓 |
Ref | Expression |
---|---|
bj-ralvw | ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 3142 | . 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} → 𝜑)) | |
2 | bj-ralvw.1 | . . . . 5 ⊢ 𝜓 | |
3 | 2 | vexw 2804 | . . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓} |
4 | 3 | a1bi 365 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} → 𝜑)) |
5 | 4 | albii 1819 | . 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: → wi 4 ↔ wb 208 ∀wal 1534 ∈ wcel 2113 {cab 2798 ∀wral 3137 |
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-sb 2069 df-clab 2799 df-ral 3142 |
This theorem is referenced by: (None) |
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