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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-abex | Structured version Visualization version GIF version |
Description: Two ways of stating that the extension of a formula is a set. (Contributed by BJ, 18-Jan-2025.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-abex | ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isset 3488 | . 2 ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦 𝑦 = {𝑥 ∣ 𝜑}) | |
2 | eqabb 2874 | . . 3 ⊢ (𝑦 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)) | |
3 | 2 | exbii 1851 | . 2 ⊢ (∃𝑦 𝑦 = {𝑥 ∣ 𝜑} ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)) |
4 | 1, 3 | bitri 275 | 1 ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wal 1540 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 Vcvv 3475 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 |
This theorem is referenced by: (None) |
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