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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-vtoclg1fv | Structured version Visualization version GIF version |
Description: Version of bj-vtoclg1f 35091 with a disjoint variable condition on 𝑥, 𝑉. This removes dependency on df-sb 2072 and df-clab 2718. Prefer its use over bj-vtoclg1f 35091 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-vtoclg1fv.nf | ⊢ Ⅎ𝑥𝜓 |
bj-vtoclg1fv.maj | ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) |
bj-vtoclg1fv.min | ⊢ 𝜑 |
Ref | Expression |
---|---|
bj-vtoclg1fv | ⊢ (𝐴 ∈ 𝑉 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elissetv 2821 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | bj-vtoclg1fv.nf | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | bj-vtoclg1fv.maj | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) | |
4 | bj-vtoclg1fv.min | . . 3 ⊢ 𝜑 | |
5 | 2, 3, 4 | bj-exlimmpi 35085 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓) |
6 | 1, 5 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∃wex 1786 Ⅎwnf 1790 ∈ wcel 2110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-12 2175 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1787 df-nf 1791 df-clel 2818 |
This theorem is referenced by: (None) |
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