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Mirrors > Home > MPE Home > Th. List > Mathboxes > elex2VD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of elex2 3216. (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 38790 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 ) | |
2 | idn2 38838 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 , 𝑥 = 𝐴 ▶ 𝑥 = 𝐴 ) | |
3 | eleq1a 2696 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) | |
4 | 1, 2, 3 | e12 38951 | . . . . 5 ⊢ ( 𝐴 ∈ 𝐵 , 𝑥 = 𝐴 ▶ 𝑥 ∈ 𝐵 ) |
5 | 4 | in2 38830 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 ▶ (𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ) |
6 | 5 | gen11 38841 | . . 3 ⊢ ( 𝐴 ∈ 𝐵 ▶ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ) |
7 | elisset 3215 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
8 | 1, 7 | e1a 38852 | . . 3 ⊢ ( 𝐴 ∈ 𝐵 ▶ ∃𝑥 𝑥 = 𝐴 ) |
9 | exim 1761 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥 𝑥 ∈ 𝐵)) | |
10 | 6, 8, 9 | e11 38913 | . 2 ⊢ ( 𝐴 ∈ 𝐵 ▶ ∃𝑥 𝑥 ∈ 𝐵 ) |
11 | 10 | in1 38787 | 1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1481 = wceq 1483 ∃wex 1704 ∈ wcel 1990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-12 2047 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-an 386 df-tru 1486 df-ex 1705 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-v 3202 df-vd1 38786 df-vd2 38794 |
This theorem is referenced by: (None) |
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