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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-vtoclf | Structured version Visualization version GIF version |
Description: Remove dependency on ax-ext 2790, df-clab 2797 and df-cleq 2811 (and df-sb 2061 and df-v 3494) from vtoclf 3556. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-vtoclf.nf | ⊢ Ⅎ𝑥𝜓 |
bj-vtoclf.s | ⊢ 𝐴 ∈ 𝑉 |
bj-vtoclf.maj | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
bj-vtoclf.min | ⊢ 𝜑 |
Ref | Expression |
---|---|
bj-vtoclf | ⊢ 𝜓 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-vtoclf.nf | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | bj-vtoclf.s | . . . . 5 ⊢ 𝐴 ∈ 𝑉 | |
3 | 2 | bj-issetiv 34090 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝐴 |
4 | bj-vtoclf.maj | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | biimpd 230 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) |
6 | 3, 5 | eximii 1828 | . . 3 ⊢ ∃𝑥(𝜑 → 𝜓) |
7 | 1, 6 | 19.36i 2223 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
8 | bj-vtoclf.min | . 2 ⊢ 𝜑 | |
9 | 7, 8 | mpg 1789 | 1 ⊢ 𝜓 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 df-clel 2890 |
This theorem is referenced by: bj-vtocl 34129 |
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