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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 2702, df-clab 2709 and df-cleq 2723 (and df-sb 2067 and df-v 3475) from vtoclf 3551. (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 36221 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝐴 |
4 | bj-vtoclf.maj | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | biimpd 228 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) |
6 | 3, 5 | eximii 1838 | . . 3 ⊢ ∃𝑥(𝜑 → 𝜓) |
7 | 1, 6 | 19.36i 2223 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
8 | bj-vtoclf.min | . 2 ⊢ 𝜑 | |
9 | 7, 8 | mpg 1798 | 1 ⊢ 𝜓 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1540 Ⅎwnf 1784 ∈ wcel 2105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-12 2170 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1781 df-nf 1785 df-clel 2809 |
This theorem is referenced by: bj-vtocl 36260 |
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