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Mirrors > Home > MPE Home > Th. List > Mathboxes > eliminable-veqab | Structured version Visualization version GIF version |
Description: A theorem used to prove the base case of the Eliminability Theorem (see section comment): variable equals abstraction. (Contributed by BJ, 30-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
eliminable-veqab | ⊢ (𝑥 = {𝑦 ∣ 𝜑} ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ [𝑧 / 𝑦]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2752 | . 2 ⊢ (𝑥 = {𝑦 ∣ 𝜑} ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝜑})) | |
2 | eliminable-velab 34569 | . . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) | |
3 | 2 | bibi2i 342 | . . 3 ⊢ ((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝜑}) ↔ (𝑧 ∈ 𝑥 ↔ [𝑧 / 𝑦]𝜑)) |
4 | 3 | albii 1822 | . 2 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝜑}) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ [𝑧 / 𝑦]𝜑)) |
5 | 1, 4 | bitri 278 | 1 ⊢ (𝑥 = {𝑦 ∣ 𝜑} ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ [𝑧 / 𝑦]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∀wal 1537 = wceq 1539 [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-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 |
This theorem is referenced by: eliminable-abelv 34573 eliminable-abelab 34574 |
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