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Mirrors > Home > MPE Home > Th. List > Mathboxes > eliminable-abelv | Structured version Visualization version GIF version |
Description: A theorem used to prove the base case of the Eliminability Theorem (see section comment): abstraction belongs to variable. (Contributed by BJ, 30-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
eliminable-abelv | ⊢ ({𝑥 ∣ 𝜑} ∈ 𝑦 ↔ ∃𝑧(∀𝑡(𝑡 ∈ 𝑧 ↔ [𝑡 / 𝑥]𝜑) ∧ 𝑧 ∈ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfclel 2832 | . 2 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝑦 ↔ ∃𝑧(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 ∈ 𝑦)) | |
2 | eliminable-veqab 34570 | . . . 4 ⊢ (𝑧 = {𝑥 ∣ 𝜑} ↔ ∀𝑡(𝑡 ∈ 𝑧 ↔ [𝑡 / 𝑥]𝜑)) | |
3 | 2 | anbi1i 627 | . . 3 ⊢ ((𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 ∈ 𝑦) ↔ (∀𝑡(𝑡 ∈ 𝑧 ↔ [𝑡 / 𝑥]𝜑) ∧ 𝑧 ∈ 𝑦)) |
4 | 3 | exbii 1850 | . 2 ⊢ (∃𝑧(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑧(∀𝑡(𝑡 ∈ 𝑧 ↔ [𝑡 / 𝑥]𝜑) ∧ 𝑧 ∈ 𝑦)) |
5 | 1, 4 | bitri 278 | 1 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝑦 ↔ ∃𝑧(∀𝑡(𝑡 ∈ 𝑧 ↔ [𝑡 / 𝑥]𝜑) ∧ 𝑧 ∈ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 400 ∀wal 1537 = wceq 1539 ∃wex 1782 [wsb 2070 ∈ wcel 2112 {cab 2736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-ex 1783 df-clab 2737 df-cleq 2751 df-clel 2831 |
This theorem is referenced by: (None) |
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