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Mirrors > Home > MPE Home > Th. List > Mathboxes > prtlem100 | Structured version Visualization version GIF version |
Description: Lemma for prter3 34167. (Contributed by Rodolfo Medina, 19-Oct-2010.) |
Ref | Expression |
---|---|
prtlem100 | ⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵 ∈ 𝑥 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anass 681 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ ∅) ∧ (𝐵 ∈ 𝑥 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ≠ ∅ ∧ (𝐵 ∈ 𝑥 ∧ 𝜑)))) | |
2 | eldifsn 4317 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ ∅)) | |
3 | 2 | anbi1i 731 | . . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ {∅}) ∧ (𝐵 ∈ 𝑥 ∧ 𝜑)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ ∅) ∧ (𝐵 ∈ 𝑥 ∧ 𝜑))) |
4 | ne0i 3921 | . . . . . . 7 ⊢ (𝐵 ∈ 𝑥 → 𝑥 ≠ ∅) | |
5 | 4 | pm4.71ri 665 | . . . . . 6 ⊢ (𝐵 ∈ 𝑥 ↔ (𝑥 ≠ ∅ ∧ 𝐵 ∈ 𝑥)) |
6 | 5 | anbi1i 731 | . . . . 5 ⊢ ((𝐵 ∈ 𝑥 ∧ 𝜑) ↔ ((𝑥 ≠ ∅ ∧ 𝐵 ∈ 𝑥) ∧ 𝜑)) |
7 | anass 681 | . . . . 5 ⊢ (((𝑥 ≠ ∅ ∧ 𝐵 ∈ 𝑥) ∧ 𝜑) ↔ (𝑥 ≠ ∅ ∧ (𝐵 ∈ 𝑥 ∧ 𝜑))) | |
8 | 6, 7 | bitri 264 | . . . 4 ⊢ ((𝐵 ∈ 𝑥 ∧ 𝜑) ↔ (𝑥 ≠ ∅ ∧ (𝐵 ∈ 𝑥 ∧ 𝜑))) |
9 | 8 | anbi2i 730 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝐵 ∈ 𝑥 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ≠ ∅ ∧ (𝐵 ∈ 𝑥 ∧ 𝜑)))) |
10 | 1, 3, 9 | 3bitr4ri 293 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝐵 ∈ 𝑥 ∧ 𝜑)) ↔ (𝑥 ∈ (𝐴 ∖ {∅}) ∧ (𝐵 ∈ 𝑥 ∧ 𝜑))) |
11 | 10 | rexbii2 3039 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵 ∈ 𝑥 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 ∖ cdif 3571 ∅c0 3915 {csn 4177 |
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-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-rex 2918 df-v 3202 df-dif 3577 df-nul 3916 df-sn 4178 |
This theorem is referenced by: (None) |
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