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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-vtoclg1f1 | Structured version Visualization version GIF version |
Description: The FOL content of vtoclg1f 3567 (hence not using ax-ext 2793, df-cleq 2814, df-nfc 2963, df-v 3497). Note the weakened "major" hypothesis and the disjoint variable condition between 𝑥 and 𝐴 (needed since the nonfreeness quantifier for classes is not available without ax-ext 2793; as a byproduct, this dispenses with ax-11 2156 and ax-13 2386). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-vtoclg1f1.nf | ⊢ Ⅎ𝑥𝜓 |
bj-vtoclg1f1.maj | ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) |
bj-vtoclg1f1.min | ⊢ 𝜑 |
Ref | Expression |
---|---|
bj-vtoclg1f1 | ⊢ (∃𝑦 𝑦 = 𝐴 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-denotes 34183 | . 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴) | |
2 | bj-vtoclg1f1.nf | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | bj-vtoclg1f1.maj | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) | |
4 | bj-vtoclg1f1.min | . . 3 ⊢ 𝜑 | |
5 | 2, 3, 4 | bj-exlimmpi 34223 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓) |
6 | 1, 5 | sylbi 219 | 1 ⊢ (∃𝑦 𝑦 = 𝐴 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∃wex 1776 Ⅎwnf 1780 |
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-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-clel 2893 |
This theorem is referenced by: (None) |
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