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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-vtoclg | Structured version Visualization version GIF version |
Description: A version of vtoclg 3538 with an additional disjoint variable condition (which is removable if we allow use of df-clab 2705, see bj-vtoclg1f 36386), which requires fewer axioms (i.e., removes dependency on ax-6 1964, ax-7 2004, ax-9 2109, ax-12 2164, ax-ext 2698, df-clab 2705, df-cleq 2719, df-v 3471). (Contributed by BJ, 2-Jul-2022.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-vtoclg.maj | ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) |
bj-vtoclg.min | ⊢ 𝜑 |
Ref | Expression |
---|---|
bj-vtoclg | ⊢ (𝐴 ∈ 𝑉 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elissetv 2809 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | bj-vtoclg.maj | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) | |
3 | bj-vtoclg.min | . . 3 ⊢ 𝜑 | |
4 | 2, 3 | bj-exlimvmpi 36379 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∃wex 1774 ∈ wcel 2099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 df-clel 2805 |
This theorem is referenced by: bj-zfauscl 36392 |
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