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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ceqsalgv | Structured version Visualization version GIF version |
Description: Version of bj-ceqsalg 34200 with a disjoint variable condition on 𝑥, 𝑉, removing dependency on df-sb 2066 and df-clab 2800. Prefer its use over bj-ceqsalg 34200 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ceqsalgv.1 | ⊢ Ⅎ𝑥𝜓 |
bj-ceqsalgv.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
bj-ceqsalgv | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-elissetv 34186 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | bj-ceqsalgv.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | bj-ceqsalgv.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | bj-ceqsalg0 34199 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1531 = wceq 1533 ∃wex 1776 Ⅎwnf 1780 ∈ wcel 2110 |
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-12 2172 |
This theorem depends on definitions: df-bi 209 df-an 399 df-3an 1085 df-ex 1777 df-nf 1781 df-clel 2893 |
This theorem is referenced by: bj-ceqsal 34204 |
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