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Mirrors > Home > MPE Home > Th. List > Mathboxes > elex2VD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of elex2 2816. (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 42846 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 ) | |
2 | idn2 42885 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 , 𝑥 = 𝐴 ▶ 𝑥 = 𝐴 ) | |
3 | eleq1a 2833 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) | |
4 | 1, 2, 3 | e12 42996 | . . . . 5 ⊢ ( 𝐴 ∈ 𝐵 , 𝑥 = 𝐴 ▶ 𝑥 ∈ 𝐵 ) |
5 | 4 | in2 42877 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 ▶ (𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ) |
6 | 5 | gen11 42888 | . . 3 ⊢ ( 𝐴 ∈ 𝐵 ▶ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ) |
7 | elisset 2819 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
8 | 1, 7 | e1a 42899 | . . 3 ⊢ ( 𝐴 ∈ 𝐵 ▶ ∃𝑥 𝑥 = 𝐴 ) |
9 | exim 1836 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥 𝑥 ∈ 𝐵)) | |
10 | 6, 8, 9 | e11 42960 | . 2 ⊢ ( 𝐴 ∈ 𝐵 ▶ ∃𝑥 𝑥 ∈ 𝐵 ) |
11 | 10 | in1 42843 | 1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 = wceq 1541 ∃wex 1781 ∈ wcel 2106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-vd1 42842 df-vd2 42850 |
This theorem is referenced by: (None) |
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