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| Mirrors > Home > MPE Home > Th. List > Mathboxes > elex2VD | Structured version Visualization version GIF version | ||
| Description: Virtual deduction proof of elex2 2818. (Contributed by Alan Sare, 25-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| elex2VD | ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | idn1 44594 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 ) | |
| 2 | idn2 44633 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 , 𝑥 = 𝐴 ▶ 𝑥 = 𝐴 ) | |
| 3 | eleq1a 2836 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) | |
| 4 | 1, 2, 3 | e12 44744 | . . . . 5 ⊢ ( 𝐴 ∈ 𝐵 , 𝑥 = 𝐴 ▶ 𝑥 ∈ 𝐵 ) | 
| 5 | 4 | in2 44625 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 ▶ (𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ) | 
| 6 | 5 | gen11 44636 | . . 3 ⊢ ( 𝐴 ∈ 𝐵 ▶ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ) | 
| 7 | elisset 2823 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
| 8 | 1, 7 | e1a 44647 | . . 3 ⊢ ( 𝐴 ∈ 𝐵 ▶ ∃𝑥 𝑥 = 𝐴 ) | 
| 9 | exim 1834 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥 𝑥 ∈ 𝐵)) | |
| 10 | 6, 8, 9 | e11 44708 | . 2 ⊢ ( 𝐴 ∈ 𝐵 ▶ ∃𝑥 𝑥 ∈ 𝐵 ) | 
| 11 | 10 | in1 44591 | 1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1538 = wceq 1540 ∃wex 1779 ∈ wcel 2108 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-vd1 44590 df-vd2 44598 | 
| This theorem is referenced by: (None) | 
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