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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 3499 not using ax-ext 2704 (nor df-cleq 2725, df-clel 2811, df-v 3477), and only core FOL axioms. See also bj-rexvw 35698. The analogues for reuv 3501 and rmov 3502 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 3063 | . 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} → 𝜑)) | |
2 | bj-ralvw.1 | . . . . 5 ⊢ 𝜓 | |
3 | 2 | vexw 2716 | . . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓} |
4 | 3 | a1bi 363 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} → 𝜑)) |
5 | 4 | albii 1822 | . 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} → 𝜑)) |
6 | 1, 5 | bitr4i 278 | 1 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 ∈ wcel 2107 {cab 2710 ∀wral 3062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-sb 2069 df-clab 2711 df-ral 3063 |
This theorem is referenced by: (None) |
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